Non-uniform specification for piecewise monotonic interval maps

1. Coding spaces for a class of interval maps

The starting point for this post is Franz Hofbauer’s paper “On intrinsic ergodicity of piecewise monotonic transformations with positive entropy”, which is actually two papers; both appeared in Israel J. Math, one in 1979 and one in 1981. He proves existence and uniqueness of the measure of maximal entropy (MME) for a map provided the following are true:

1. there are disjoint intervals such that and is continuous and strictly monotonic;
2. the partition into these intervals is generating — that is, is dense in , where denotes the set of all endpoints of the intervals ;
3. ;
4. is topologically transitive.

Hofbauer’s proof proceeds by modeling using a countable-state Markov shift, and actually gives quite a bit of information even without topological transitivity, including a decomposition of the non-wandering set into transitive pieces, each of which supports at most one MME.

I want to describe an approach to these maps using non-uniform specification. These maps admit a natural symbolic coding on the finite alphabet obtained by tracking which interval an orbit lands in at each iterate, and this is the lens through which we will study them.

To make this more precise, suppose that satisfies Condition 1 (piecewise monotonicity), and let denote the set of all words such that the following interval is non-empty:

The letter “I” can be understood as standing for “interval” or for “initial”, in the sense that an orbit segment is coded by the word if and only if its initial point lies in . In this case we have

where the “F” in this notation can be understood as standing for “following” (or “final” if you prefer). Observe that the intervals are precisely the intervals on which is continuous and monotonic, and thus each is an interval as well.

Exercise 1 Suppose that is uniformly expanding on each in the sense that there is such that whenever , and prove that in this case Condition 2 (generating) is satisfied.

Exercise 2 Suppose that satisfies Conditions 1 and 2. Show that is the language of a shift space , and that there is a semi-conjugacy defined by

(The fact that this intersection is a single point relies on Condition 2.) Show that there is a countable set such that is 1-1.

Exercise 3 Show that if is uniformly expanding on each , then Condition 3 (positive entropy) is satisfied, and in particular has positive topological entropy .

Under Conditions 1–3, there is a 1-1 correspondence between positive entropy invariant measures for and positive entropy invariant measures for , because the countable set can support only zero entropy measures. Thus to study the MMEs for , it suffices to study the MMEs for the shift space .

In general one cannot expect this shift space to be Markov. However, it turns out that in many cases it does have the non-uniform specification property described in my last post, which will lead to an alternate proof of intrinsic ergodicity (that is, existence of a unique MME). One of my motivations for writing out the details of this here is the hope that these techniques might eventually be applicable to more general hyperbolic systems with singularities, in particular to billiard systems; piecewise expanding interval maps provide a natural sandbox in which to experiment with the approach before grappling with the difficulties introduced by the presence of a stable direction.

To this end, the goal for this post is to exhibit collections of words with the following properties.

• Decomposition: For every , there are and such that .
• Specification: There is such that for every , there is with such that .
• Entropy gap: .

Once these are proved, the main theorem from the previous post guarantees existence of a unique MME. (The specification property there requires , but this was merely to simplify the exposition; with a little more work one can establish the result under the weaker condition here.)

2. Exploring -transformations

Rather than stating a definitive result immediately, let us go step-by-step — in particular, we will eventually add another condition on (beyond Conditions 1–3) to replace topological transitivity, and the initial definition of and will only cover certain examples, with a modification required to handle the general case.

Start by considering the -transformation when , whose first three iterates are shown in Figure 2. Labeling the branches with the alphabet , we see that and ; in the figure, these intervals are marked on both the horizontal and vertical axes. Clearly , and for this specific choice of we have , so .

Looking at the second picture in the figure, which shows the graph of , we can see that

The third picture shows a similar phenomenon for , with when , and when .

Exercise 4 For this particular map , show by induction that if ends in a , and if ends in a . In particular, we have for all . Use this to deduce that consists of precisely those words that do not contain as a subword. (Thus is a transitive SFT and has a unique MME.)

The key step in this exercise is the observation that given and such that , the definition of the intervals and in (1) and (2) gives

and thus

In particular, if , then we have , and thus

A similar argument replacing with a word (and the first iterate with ) gives the following lemma, whose detailed proof is left as an exercise.

Lemma 1 Given any map satisfying Conditions 1 and 2, and any , we have

In particular, if , then and .

Now consider the map for ; Figure 2 shows the first three iterates. Observe that we once again have and , so that the same argument as in Exercise 4 shows that if is any word that does not contain as a subword, then . However, this is no longer a complete description of the language, since as the second picture in Figure 2 illustrates, we have (equivalently, , as seen in the first picture) — consequently .

In Exercise 4, every word had the property that ; writing for and , we can rewrite this property as , which by Lemma 1 occurs if and only if . These two properties can be seen on Figure 2 as follows.

• : the vertical interval spanned by the graph of on is the same as the vertical interval spanned by the graph of on .
• :
  on the horizontal interval , the graph of crosses the vertical interval completely.

Looking at Figure 2, we see that now these properties are satisfied for some words, but not for all. Let us write

Exercise 5 Referring to Figure 2, prove that and that . In particular, , , and lie in but not in . Show that on the other hand, (at least for the value of illustrated), so that words in can have as a subword.

The notation suggests that we are going to try to prove the specification property for this collection of words. The first step in this direction is to prove the following Markov-type property.

Lemma 2 Let satisfy Conditions 1 and 2, and let be given by (4). If have the property that , then , where (concatenation with the repeated symbol only appearing once).

Proof: Write where and , so that . By Lemma 1, we have

Since and , we have , so and we conclude that

where the last equality uses the fact that . This shows that .

In the specific case of a -transformation with , we can refer to Figure 2 and Exercise 5 to see that contains the words , , , and . In particular, for any , we can choose to be the one of these four words that begins with the last symbol in , and ends with the first symbol in . Then and ; applying Lemma 2 twice gives , which proves the specification property.

3. Obtaining a specification property

In the general case of an interval map satisfying Conditions 1 and 2, we would like to play the same game, proving specification for by mimicking the proof of specification for a topologically transitive SFT: given , we would like to let be such that and ; then given any , we can put and apply Lemma 2 twice to get . Since is finite, this would prove that has specification with .

But how can we guarantee that such a exists for every ? This does not follow from Conditions 1–3, and indeed one can easily produce examples satisfying these conditions for which there are multiple MMEs and the collection defined in (4) does not have specification, because there are some pairs for which no such exists.

At this point it is tempting to simply add existence of the required words as an extra condition, replacing topological transitivity. But it is worth working a little harder. Consider the map , three iterates of which are illustrated in Figure 3 for and .

For this , we see from the first picture that neither of or contains , and thus there can be no word such that , since we always have . It follows that does not contain any words ending in , which rules out a proof of specification along the lines above.

The story is different if instead of single letters, we look at words of length two. Observe from the second picture that

This suggests that we should modify the definition of in (4) to require that the follower set depends only on the last two symbols (instead of the last one). More generally, we can fix and define

We have the following generalization of Lemma 2.

Lemma 3 If (with ) have the property that , then .

Proof: Using Lemma 1, we have

and since , we have , so , giving

since . This shows that .

Now we can prove that has specification for the map in Figure 3; as described above, it suffices to prove that if are any two words from the set , then there is some . But this follows quickly from the fact that the transitions are all allowed by (5), and that any word obtained by concatenating allowable transitions must lie in by iterating Lemma 3.

This discussion motivates the following.

Definition 4 Given , define a directed graph whose vertices are the words in and whose edges are given by the condition that exactly when . Say that has Property if this graph is strongly connected, meaning that given any two , there is a path in the graph from to .

Property is playing the role of topological transitivity, but the exact relationship is not quite clear yet. I’ll have more to say about this in my next post, and the formulation here is almost certainly not the “optimal” one, but since this is a blog and not a paper, let’s stick with this property for the time being so that we can formulate an actual result and not get lost in the weeds.

Proposition 5 Let satisfy Conditions 1 and 2, and Property for some . Then has specification.

Proof: By Property , for every , there are such that . Thus . Applying Lemma 3 repeatedly shows that . Write , so .

Now let . Given any with , let , so that . One application of Lemma 3 gives , and a second application gives . Since , this completes the proof.

4. Bounding the entropy of suffixes

Let be an interval map satisfying Conditions 1–3 and Property . Proposition 5 shows that has specification, so it remains to find such that the decomposition and entropy gap conditions are satisfied.

For simplicity, let us first discuss the case . Given , we want to write where and lies in some yet-to-be-determined collection . Since this collection should be “small”, it makes sense to ask for to be as long as possible, and thus it is reasonable to take , where is the largest index in with the property that

$latex \displaystyle

w_1\cdots w_m \in {\mathcal G}. \ \ \ \ \ (7)&fg=000000$

In particular, for every , we have

$latex \displaystyle

w_1 \cdots w_\ell \notin {\mathcal G}. \ \ \ \ \ (8)&fg=000000$

By Lemma 2, this implies that : indeed, if we had , then using (7) and Lemma 2 would give , contradicting (8).

This shows that if we put , then lies in the collection

$latex \displaystyle

{\mathcal C}(a) := \{ v\in {\mathcal L} : av\in {\mathcal L}, av_1\cdots v_\ell \notin {\mathcal G} \ \forall 1\leq \ell \leq |v| \}.

\ \ \ \ \ (9)&fg=000000$

Taking gives the desired decomposition. But what is the entropy of ? First note that if

for some and , then , so we can bound in terms of by controlling how many choices of there are for each .

The key is in the observation that and are intervals that intersect in more than one point (since ), but that (since ). The only way for this to happen is

if the interior of contains one of the endpoints of . Since the interiors of the intervals are disjoint, we see that for any there are at most two choices of for which . It follows that

and iterating this gives

We conclude that

If , then this provides the needed entropy gap, and we are done.

But what if lies in ? This is where we need to use instead of . Given any and , let

$latex \displaystyle

{\mathcal C}^{(k)}(q) := \{v \in {\mathcal L} : qv\in {\mathcal L}, q v_1 \cdots v_\ell \notin {\mathcal G}^{(k)} \ \forall 1\leq \ell \leq |v| \}.

\ \ \ \ \ (10)&fg=000000$

Following the same idea as in the case , we see that every word in is of the form , where and are such that the intervals and intersect in more than one point (since ) but (since ). The only way for this to happen is if the interior of contains one of the endpoints of . Since the interiors of the intervals are disjoint (as ranges over ), we see that for any there are at most two choices of for which . It follows that

and iterating this gives

We conclude that

as long as we choose .

Together with Proposition 5, this proves the following.

Theorem 6 Let be an interval map satisfying Conditions 1–3 and Property for some . Then the coding space for has a language with a decomposition satisfying the specification and entropy gap conditions; in particular, has a unique MME.

Some natural questions present themselves at this point.

• How does Property relate to the topological transitivity condition from Hofbauer’s paper?
• Does having Property for some value of tell us anything about having it for other values? (If so, one might hope to be able to remove the condition on in Theorem 6.)
• Is there a more general version of Property that we can use in Proposition 5 and Theorem 6?

I’ll address these in the next post. For now I will conclude this post by emphasizing the idea in the proof of (11): “if we go iterates in each step, then since each step only has two ways to be bad, we can bound the entropy of always-bad things by ”. This same idea plays a key role in the proof of the growth lemma for dispersing billiards (although in that setting one needs to replace “two ways” with “ ways” and the bound becomes ); see Chapter 5 of the book “Chaotic Billiards” by Chernov and Markarian, or Section 5 of “On the measure of maximal entropy for finite horizon Sinai Billard maps” by Baladi and Demers (JAMS, 2020) (in particular, compare equation (5.4) there with (11) above). The fact that taking large makes this estimate smaller than the entropy is an example of the phenomenon “growth beats fragmentation”, which is important in studying systems with singularities.

An improved non-uniform specification result

My last post (nearly two years ago!) described Bowen’s proof that a shift space with the specification property has a unique measure of maximal entropy. A little over ten years ago, Dan Thompson and I gave a version of Bowen’s proof that turned out to be well-suited for applications to non-uniformly hyperbolic systems; an overview of the approach and of various extensions and applications since then, especially to equilibrium states for geodesic flows, can be found in our survey article in “Thermodynamic Formalism: CIRM Jean-Morlet Chair, Fall 2019” (Springer LNM, volume 2290). Today I want to return to the original setting of shift spaces, as described in our original paper (Israel J. Math, 2012), which I will refer to as [CT] from now on.

Our main result in [CT] contained an assumption that always bothered me (Condition (III) in Theorem C, see below for a description). I really had the feeling that the result should remain true even without it, but we couldn’t figure out how to remove it. Recently, Maria Jose Pacifico, Fan Yang, and Jiagang Yang have managed to remove this hypothesis in a new paper, which I will refer to as [PYY]. They consider flows rather than shift spaces, and it turns out that for flows with fixed points, including the Lorenz attractors that they are interested in, verifying the “extra assumption” (a version of which does indeed appear in my 2016 paper with Dan Thompson on non-uniform specification for flows) actually presents real problems, so their ability to prove a uniqueness result without requiring this condition has not just aesthetic but also practical value.

The arguments for flows are substantially more technical than those for shift spaces, and the proofs involve a steady stream of Bowen balls, adapted partitions, separated sets, two-scale partition sums, bounded distortion estimates, etc., which somewhat obscure the core innovation in [PYY] that brings the improvement over [CT]. The purpose of this post is to translate the argument from [PYY] back to the symbolic setting in order to clarify exactly where this improvement lies; the results I describe below can all be formally deduced from [PYY], but I will give complete proofs of everything that does not appear already in [CT].

Basic notation for shift spaces, etc., will be as in my previous post. To formulate the (improved) uniqueness result, we need the notion from [CT] of a decomposition of the language of a shift space : this is a choice of three collections of words such that every can be decomposed as a `prefix’ from , followed by a `good core’ from , followed by a `suffix’ from . Formally, for every , there are , , and such that . See [CT] for examples of shift spaces with decompositions satisfying the conditions in the following theorem. (Spoiler: -shifts and -gap shifts.)

Theorem 1 Suppose is a shift space on a finite alphabet with topological entropy whose language has a decomposition satisfying the following conditions.

(I) Specification for . There is such that for all , there are such that .

(II) Entropy gap for . .

Then has a unique MME.

In the proofs (both in [CT] and in [PYY]) one must work not just with but also with the collections

Observe that ; if you are familiar with Pesin theory for non-uniformly hyperbolic diffeomorphisms, it is reasonable to think of this as an analogue of the decomposition of a (non-uniformly) hyperbolic set into regular level sets. (If you are not familiar with Pesin theory, you should probably just ignore that last sentence.) In [CT], we required the following extra condition.

(III) For every , there exists such that given , there exist words with length such that .

This condition guarantees that each of the collections has (a slightly weaker version of) specification as well, with gap size allowed to depend on , and it appears in this form in all of our later papers; this is the condition that turns out to present problems for Lorenz attractors, and indeed for general flows with fixed points, and this is what motivated the authors of [PYY].

Let me mention in passing that there is also a symbolic setting where (III) becomes problematic; a few years ago I wrote a paper with Ronnie Pavlov (ETDS, 2019) where we studied shift spaces with “one-sided almost specification with bounded mistake function” and proved uniqueness of the MME. We were able to verify conditions (I) and (II) for these examples, but not condition (III) (see the end of §2.3 in that paper), and ultimately needed to use a slightly different set of conditions I had formulated in a paper constructing countable-state Markov models for shifts with non-uniform specification (Comm. Math. Phys., 2018). These conditions are a little more complicated, and the resulting proofs are a lot more complicated, so it is nice to know now that the uniqueness results in my paper with Ronnie Pavlov do in fact follow from the simpler and shorter proof of Theorem 1.

To see where [PYY] makes the improvement leading to a proof of Theorem 1 without requiring (III), start by recalling the overall structure of Bowen’s proof as I outlined it in my last post: first one uses specification to construct a Gibbs measure , then one proves that is ergodic and that there can be no mutually singular MME . The proof in [CT] follows similar lines, and roughly speaking, Condition (III) was only required for the second half. More precisely, using only Conditions (I) and (II), [CT] still contains a proof of the following.

Proposition 2 There exists an MME that has the following Gibbs-type property: there is such that

Moreover, there is such that

Proof: See Lemmas 5.1–5.10 in [CT] for the proofs of the uniform counting bounds and the construction of a measure satisfying (1); note the sentence after the proof of Lemma 5.10 emphasizing that Condition (III) was not used. The proof that this measure also satisfies the “two-step Gibbs bound” (2) is contained in the proof of [CT, Lemma 5.15], even though the statement of that lemma is a little different. (In fact one can take to be arbitrarily large, going to infinity along a sequence with bounded gaps, but we only need (2) for a single value of .)

The remainder of the proof in [CT] uses (2) to establish ergodicity and (1) to rule out the existence of a mutually singular MME. More precisely, this is done by using the fact under Condition (III), Proposition 2 still holds with replaced by (although the constant is allowed to depend on ). One also uses the fact that various sets that appear in the arguments can be well-approximated by unions of cylinders corresponding to words in .

Without Condition (III), there is no guarantee that Proposition 2 holds for , and so instead [PYY] must work with the “good cores” of words, and must approximate the relevant sets with unions of cylinders from , not . The two lemmas that make this precise both require a certain `truncation operation’ to state properly: given and with , let

denote the word obtained by deleting the first and last symbols from . Given a collection of words , we will write for the union of the corresponding cylinders.

Lemma 3 There exist and such that if is any MME and satisfies , then there are (depending on and ) such that .

(In fact one can replace with any , at the cost of possibly needing to use a different and .)

Proof: This very nearly follows from two lemmas in [CT]:

• Lemma 5.7: For all there exists such that if is any MME and , then .
• Lemma 5.6: For every , there exists such that if , then . (The formulation in [CT] is slightly different, but the proof there gives this statement.)

Combining these gives such that

The remaining step is not done in [CT] but is done in [PYY, Lemma 5.2]. Observe that for every there are such that ; moreover, for any , there are at most words with . It follows that

Combining this with (3) gives

which proves Lemma 3 with .

From now on, we fix and as given by Lemma 3. The key innovation for approximating sets using (instead of ) is in [PYY, Lemma 7.3], which implies the following in our simple setting.

Lemma 4 Given any mutually disjoint invariant Borel probability measures on , and any , there is a collection of words and such that

1. for all , and
2. for all satisfying and , in particular, for all .

Proof: Since and both are invariant, there are disjoint invariant sets such that for . By inner regularity there are compact sets such that .

(If one only wanted the results to hold for , then one could at this point put and be done; this is a fairly standard argument and is what we used in [CT], where in fact we did not even bother to spell it out so explicitly. The extension to general — which is needed in order to leverage Lemma 3, as we will see later — requires a little more care in the choice of , and this is where [PYY] is quite clever in strengthening the usual argument.)

Now given , let . It follows immediately that , and since is invariant we get

which proves the first claim. For the second claim, we take large enough that no -cylinder intersects both and (this is possible since these are compact disjoint sets). Then for every as in the statement, and every , the word has , which intersects , so , and moreover , so ; that is, . We conclude that

Taking a union over all gives , and now invariance of gives

which completes the proof of Lemma 4.

With Lemmas 3 and 4 in hand, we can now follow [PYY] in completing the second part of the uniqueness argument by showing that there is no MME , and that is ergodic. The first of these follows [PYY, Proposition 7.2]. Recall that is the constant in the lower Gibbs bounds (1) and (2), and are given by Lemma 3.

Proposition 5 There is no MME .

Proof: Suppose for a contradiction that there is another MME . Fix , and apply Lemma 4 with and to get and such that for all , and

for all satisfying .

Since , we have for all , and since is an MME, Lemma 3 gives (depending on ) with

thus the Gibbs property on gives

For all and this choice of , we also have (4), which leads to

This contradicts our choice of and completes the proof of Proposition 5.

The proof of ergodicity follows [PYY, Proposition 7.4].

Proposition 6 is ergodic.

Proof: Suppose for a contradiction that is invariant and . Let and be the normalizations of restricted to and , respectively. Fix , and apply Lemma 4 twice — once with and , and once with the roles reversed — to obtain and such that and for every , and moreover

for all satisfying .

Observe that each of and is an MME (since entropy depends affinely on the measure, and the MME is a convex combination of and ). Thus we can apply Lemma 3 to both and , obtaining (depending on ) such that the collections

both have cardinality at least .

Let , so that for every . Given any such , and any , the “two-step” Gibbs property in (2) gives

Summing over all such and gives

We will get an upper bound on this measure and derive a contradiction. Given any , we either have or (since is invariant), and thus

Observe that whenever , (5) gives

and similarly

so that

Comparing this with (6) gives , contradicting our choice of and completing the proof of Proposition 6.

Specification and the measure of maximal entropy

These are notes for a talk I am giving in Jon Chaika’s online working seminar in ergodic theory. The purpose of the talk is to outline Bowen’s proof of uniqueness of the measure of maximal entropy for shift spaces with the specification property. Bowen’s approach extends much more broadly than this: to non-symbolic systems (assuming expansivity); to equilibrium states for non-zero potential functions (assuming a bounded distortion property); and to non-uniformly hyperbolic systems using the notion of “obstructions to specification and expansivity” developed by Dan Thompson and myself (see some notes here and here, and videos here). In these notes, though, I want to give the bare bones of the argument in the simplest possible setting, to make the essential structure as clear as possible. I gave an alternate argument in a previous post; here I am giving Bowen’s original argument, although I do not necessarily follow his presentation. I should also point out that this argument differs from the construction of the MME, or more generally the equilibrium state, in Bowen’s monograph, which uses the Ruelle operator.

1. Setting and result

Let be a finite set; the alphabet. Then the full shift is the set of infinite sequences of symbols from ; this is a compact metric space with , where . The shift map is defined by . A shift space is a closed set with .

Example 1 The best example to keep in mind through this talk is a topological Markov shift: fix and put ; then fix a transition matrix with entries in , and write

Define for all . This is a topological Markov shift (TMS). It can be viewed in terms of a directed graph with vertex set and edges given by (1): consists of all sequences that label infinite paths on the graph.

The TMS is mixing or primitive if there is such that for all . Equivalently, the graph is strongly connected and the set of loop lengths on the graph has gcd .

Given a shift space , consider

The set of invariant measures is extremely large for a mixing TMS (and more generally for systems with some sort of hyperbolic behavior), and it is important to identify “distinguished” invariant measures. One way of doing this is via the variational principle

The next section recalls the definitions of the topological and measure-theoretic entropies in this setting. A measure achieving the supremum is a measure of maximal entropy (MME).

We will see that every mixing TMS has a unique MME, via a more general result. Given and , let be the set of sequences in that start with the word (juxtaposition denotes concatenation); call this the cylinder of . Define the language of by

Definition 1 has specification if there is such that for all there is such that .

Exercise 1 Prove that every mixing TMS has specification.

Theorem 2 (Bowen) If has specification, then it has a unique MME.

2. Entropies

2.1. Topological entropy

Every word in is of the form for some and ; thus

This means that the sequence has the sub-additivity property

Exercise 2 Prove Fekete’s lemma: for any sequence satisfying (2), exists and is equal to (a priori it could be ).

We conclude that the topological entropy exists for every shift space. This quantifies the growth rate of the total complexity of the system.

Exercise 3 Prove that is the box dimension of (easy). Then prove that it is also the Hausdorff dimension of (harder). (Both of these facts rely very strongly on the fact that is globally constant whenever are close; for general systems where the amount of expansion may vary, the definition of topological entropy is more involved and the relationship to dimension is more subtle, although it is worth noting that a 1973 paper of Bowen in TAMS gives a definition of topological entropy that is analogous to Hausdorff dimension.)

2.2. Measure-theoretic entropy

Recall the motivation from information theory for the definition of entropy: given , let . This can be interpreted as the information associated to an event with probability ; note that it is monotonic (the less likely an event is, the more information we gain by learning that it happened) and that .

Now define on by (and ); this can be interpreted as the expected amount of information associated to an event with probability .

Exercise 4 Show that is concave.

Given , let and be the set of sub-probability vectors with components. Define

this can be interpreted as the expected information associated to a collection of mutually exclusive events with probabilities .

Exercise 5 Show that for all , with equality if and only if for all .

Given , we have for each a probability vector with components; writing for convenience, the entropy (expected information) associated to this vector is

Lemma 3

Proof:

where the first line is by definition, the second is since , the third uses concavity of , and the fourth uses invariance of to get .

This lemma has the following intuitive interpretation: the expected information from the first symbols is at most the expected information from the first symbols plus the expected information from the next symbols.

Now Fekete’s lemma implies that the following measure-theoretic entropy exists:

2.3. Variational principle

Using Exercise 5 we see that , with equality if and only if for all . This immediately proves that

and suggests that in order to have equality we should look for a measure with

The following makes this more precise.

Definition 4 is a Gibbs measure if there are such that for all and , we have .

Now Theorem 2 is a consequence of the following two results, which we prove below.

Theorem 5 If is an ergodic Gibbs measure for , then and is the unique MME.

Theorem 6 If has specification, then it has an ergodic Gibbs measure.

In fact the construction of below always gives equality in (6), without relying on the specification property (or obtaining uniqueness), but we will not prove this.

2.4. Convex combinations

Before embarking on the proof of Theorems 5 and 6, we establish a general property of entropy that will be important.

Lemma 7 Given and with , we have

Proof: The first inequality follows immediately from concavity of . For the second inequality we first observe that

Then we use monotonicity of to get

where the last equality uses (9) twice; this proves (8).

Applying Lemma 7 to the probability vectors associated to two measures , we see that

for all : dividing by and sending gives

Remark 1 It is worth mentioning the deeper fact that a version of (10) holds for infinite convex combinations (even uncountable ones given by an integral); this is due to Konrad Jacobs, see Section 9.6 of “Foundations of Ergodic Theory” by Viana and Oliveira.

3. Gibbs implies uniqueness

To prove Theorem 5, start by observing that the lower Gibbs bound gives

and thus . Meanwhile, the upper Gibbs bound gives

and thus , so we conclude that . It remains to show that every with has .

First observe that by (10), we can restrict our attention to the case when . Indeed, given any , the Lebesgue decomposition theorem gives for some and , and (10) gives . By ergodicity we must have and thus , so if then the same is true of .

Now consider . Then there is a Borel set with and , and this in turn gives such that

Let denote the normalization of after restricting to words in , and similarly for . Recall from Fekete’s lemma and subadditivity of that for all . Then we get

where the second line uses Lemma 7, the third line uses Exercise 5, and the fourth line uses the lower Gibbs bound as in (11). We conclude that

and as the right-hand side goes to by (12), which implies that , completing the proof.

4. Specification implies Gibbs

Now we outline the proof of Theorem 6. This comes in three steps: (1) uniform counting bounds; (2) construction of a Gibbs measure; (3) proof of ergodicity.

4.1. Uniform counting bounds

From now on we write for convenience. Fekete’s lemma gives for all , or equivalently . This can also be deduced by writing

and sending so that the left-hand side goes to (this is basically part of the proof of Fekete’s lemma).

To get an upper bound on we need the specification property, which gives a 1-1 map , so that . Then one can either apply Fekete’s lemma to , or observe that

Sending the left-hand side goes to , and so combined with the lower bound above we get the uniform counting bounds

4.2. A Gibbs measure

There is a standard procedure for constructing an MME: let be any sequence of (not necessarily invariant) Borel probability measures such that for all , and then put

Since is weak* compact, there is a weak* convergent subsequence .

Exercise 6 Show that is -invariant ().

The preceding exercise is basically the proof of the Krylov-Bogolyubov theorem, and does not require any properties of the measures beyond the fact that they are Borel probability measures.

One can prove that is an MME whether or not has specification, but we will use specification to directly prove the stronger Gibbs property.

Given any and any , we bound by estimating how many words in are of the form for some and . Arguments similar to those in the uniform counting bounds show that

where the first inequality requires specification. Dividing by and using the uniform counting bounds (13) gives

using a similar estimate from below we get

Averaging over all and sending along the subsequence gives

so is a Gibbs measure.

4.3. Ergodicity

To prove that is ergodic, start by fixing , with lengths and , respectively. Given and , follow the same procedure as above to estimate the number of words in with the form , where , , and , and obtain the bounds

Averaging over , sending , and using the uniform counting bounds (13) gives

Using the Gibbs bounds (15) gives

Exercise 7 Given Borel sets , approximate and with cylinders and use (16) to get

Now if is invariant, then taking and in Exercise 7, the lower bound gives , so is either or . This proves ergodicity and completes the proof of Theorem 6.

Remark 2 In fact, the upper bound in Exercise 7 can be used to show that is mixing; see Proposition 20.3.6 in Katok and Hasselblatt.

Cohomologous functions and the Livsic theorem

Given a dynamical system (typically is a compact metric space and is continuous), we commonly encounter real-valued functions in one of the following two roles.

1. An observable function represents a measurement of the system, so that the sequence of functions represents measurements made at different times, about which we want to make predictions. In the cases we are interested in, these predictions will be probabilistic and will be in terms of the Birkhoff averages , where is the th Birkhoff sum.
2. A potential function is used to assign weights to different trajectories of the system for purposes of selecting an invariant measure with specific dynamical or geometric properties. Generally the orbit segment is assigned a weight given by . A potential function also assigns weights to invariant measures by integration; in particular, we can study equilibrium measures for , which are invariant probability measures maximizing . (Here is Kolmogorov–Sinai entropy.)

Before moving on we remark that can also play the role of a density function, especially if we are looking for an invariant measure that is absolutely continuous with respect to some reference measure, but for today we will focus on the two roles described above.

Let denote the set of -invariant Borel probability measures on . Then each induces a map by as in the second item above. It is natural to ask when two functions have . One immediate observation to make is that the defintion of -invariance implies that for all and , so that in particular we have . Thus the function has . We call such a function a coboundary; we have just shown that

Proposition 1 if is a coboundary, then it integrates to with respect to any invariant measure.

Moreover, since integration is linear in , we have

Proposition 2 if two functions and differ by a coboundary, then , ie., for every .

If and differ by a coboundary, we say that they are cohomologous; in this case we can write for some , which we call the transfer function. Note that is a coboundary if and only if it is cohomologous to the zero function.

Remark 1 For a discussion of how this is connected to the notion of cohomology in algebraic topology, see Terry Tao’s blog post from December 2008.

It is natural to ask whether cohomology is the only mechanism by which two functions can have ; in other words, do the converses of Propositions 1 and 2 hold?

In general the answer is no, as one can quickly see by letting be an irrational circle rotation, so that the only invariant measure is Lebesgue, and there are many continuous functions on the circle that have Lebesgue integral but are not coboundaries. When has hyperbolic behavior, however, the story is quite different, and this is what we will discuss in this post.

If is a diffeomorphism and is a topologically transitive locally maximal hyperbolic set, then satisfies the following result.

Theorem 3 (Closing Lemma) For every there exists such that if and are such that , then there exists such that and for all .

Moreover, in this case every Hölder continuous has the following property, introduced by Walters in 1978.

Theorem 4 (Walters Property) For every there exist such that if and are such that for all , then .

Both the Closing Lemma and the Walters Property also hold when is a subshift of finite type and is Hölder continuous.

Using these properties we can formulate and prove an important result.

Theorem 5 (Livsic Theorem) Let be a compact metric space, a continuous map satisfying the Closing Lemma and possessing a point whose orbit is dense, and a continuous function satisfying the Walters Property. Then is a coboundary if and only if for every periodic point , we have .

The forward implication is immediate: if is a coboundary, then for every invariant , in particular for . Equivalently one can observe that Birkhoff sums of coboundaries have the following behavior: if , then

and consequently whenever . We must prove the reverse implication, that if whenever , then is a coboundary. To this end note that if is a continuous (hence uniformly continuous) transfer function for , then (1) immediately determines along the entire forward orbit of a point by

(A similar procedure defines along the backward orbit if is invertible.) If is a point whose orbit is dense in , then this determines on all of by continuity. Thus it suffices to prove that (2) defines a uniformly continuous function on the orbit of . To this end, given , let be given by the Walters Property, and then let be given by the Closing Lemma. Suppose that are two points on the orbit of such that . Since lie on the same orbit, there is such that or . Without loss of generality we assume the first case, so that . By the Closing Lemma there is a periodic point such that for all . By (2) and the Walters Property, this implies that

where the last equality uses the fact that Birkhoff sums of vanish around periodic orbits. This proves uniform continuity of on the orbit of , so extends uniformly continuously to , and it is an easy exercise using continuity of both sides of the equation to verify that this relationship continues to hold on all of , so that is indeed a coboundary. This completes the proof of the Livsic Theorem.

Let be as in the Livsic Theorem; then for any satisfying the Walters Property, the following are equivalent:

1. for some ;
2. for all ;
3. for all periodic orbit measures.

Here is one final remark on a specific example of cohomologous potentials: one immediately sees that for every , and so under the conditions of the Livsic Theorem one can conclude that each Birkhoff average is cohomologous to . In fact this can be shown directly, without using the Livsic Theorem, by putting

Then we have

which demonstrates that and are cohomologous.

Gibbs measures have local product structure

Let be a compact smooth manifold and a transitive Anosov diffeomorphism. If is an -invariant Borel probability measure on that is absolutely continuous with respect to volume, then the Hopf argument can be used to show that is ergodic. In fact, recently it has been shown that even stronger ergodic properties such as multiple mixing can be deduced; for example, see Coudène, Hasselblatt, and Troubetzkoy [Stoch. Dyn. 16 (2016), no. 2].

A more general class of measures with strong ergodic properties is given by the theory of thermodynamic formalism developed in the 1970s: given any Hölder continuous potential , the quantity is maximized by a unique invariant Borel probability measure , which is called the equilibrium state of ; the maximum value of the given quantity is the topological pressure . The unique equilibrium state has the Gibbs property: for every there is a constant such that

where is the Bowen ball around of order and radius , and we write for the th ergodic sum along the orbit of .

Historically, strong ergodic properties (mixing, K, Bernoulli) for equilibrium states have been established using methods such as Markov partitions rather than via the Hopf argument. However, in the more general non-uniformly hyperbolic setting, it can be difficult to extend these symbolic arguments, and so it is interesting to ask whether the Hopf argument can be applied instead, even if it only recovers some of the strong ergodic properties. The key property of absolutely continuous measures that is needed for the Hopf argument is the fact that they have a local product structure, which we define below. It was shown by Haydn [Random Comput. Dynam. 2 (1994), no. 1, 79–96] and by Leplaideur [Trans. Amer. Math. Soc. 352 (2000), no. 4, 1889–1912] that in the uniformly hyperbolic setting, the equilibrium states have local product structure when is Hölder continuous; thus one could apply the Hopf argument to them.

This post contains a direct proof that any measure with the Gibbs property (1) has local product structure; see Theorem 3 below. (This will be a bit of a longer post, since we need to recall several different concepts and then do some non-trivial technical work.) Since Bowen’s proof of uniqueness of equilibrium states using specification [Math. Systems Theory 8 (1974/1975), no. 3, 193–202] establishes the Gibbs property, this means that the equilibrium states produced this way could be addressed with the Hopf argument (I haven’t carried out the details yet, so I claim no formal results here). I should point out, though, that even without the use of Markov partitions, Ledrappier showed that these measures have the K property, which in particular implies multiple mixing. Since multiple mixing is the strongest thing we might hope to get from the Hopf argument, my primary motivation for the present approach is that Dan Thompson and I recently generalized Bowen’s result to systems satisfying a certain non-uniform specification property [Adv. Math. 303 (2016), 745–799], and the unique equilibrium states we obtain satisfy a non-uniform version of the Gibbs property (1), so it is reasonable to hope that they also have local product structure and can be studied using the Hopf argument; but this is beyond the scope of this post and will be addressed in a later paper.

1. Local product structure

Before defining local product structure for , we recall some definitions. Since is Anosov, every has local stable and unstable manifolds , which have the following properties.

1. There are and such that for all , , and , we have ; a similar contraction bound holds going backwards in time when .
2. There is such that if for all , then ; similarly for with .
3. There is such that if , then is a single point, which we denote . Moreover, there is a constant such that , and similarly for .

A set is a rectangle if it has diameter and is closed under the bracket operation: in other words, for every , the intersection point exists and is contained in .

Lemma 1 For every , there is a rectangle containing .

Proof: Write and similarly for . Consider the set and observe that for every we have

Thus has diameter , and for every we have

so is indeed a rectangle.

Given and a rectangle , let . Then can be recovered from via the method in the proof above, as the image of the map . Given measures on , let be the pushforward of under this map; that is, for every pair of Borel sets and , we put .

Definition 2 A measure has local product structure with respect to if for every and rectangle there are measures on such that .

Theorem 3 Let be a Anosov diffeomorphism, a Hölder continuous function, and an -invariant Borel probability measure on satisfying the Gibbs property (1) for some . Then has local product structure in the sense of Definition 2. Moreover, there is such that for all , the measures can be chosen so that the Radon–Nikodym derivative satisfies at -a.e. point.

A quick side remark: here the diffeomorphism is only required to be . The reason for the hypothesis at the beginning of this post was so that the geometric potential is Hölder continuous and its unique equilibrium state (the absolutely continuous invariant measure if it exists, or more generally the SRB measure) has the Gibbs property; this may not be true if is only .

2. Conditional measures

In order to prove Theorem 3, we must start by recalling the notion of conditional measures; see Coudène’s book (especially Chapters 14 and 15) or Viana’s notes for more details than what is provided here.

Let be a Lebesgue space. A partition of is a map such that for -a.e. , the sets and either coincide or are disjoint. Write for the set of partition elements, and say that the partition is finite if is finite.

Given a finite partition , it is easy to define conditional measures on the set for -a.e. by writing

when , and ignoring those partition elements with zero measure. One can recover the measure from its conditional measures by the formula

If we write for the measure on defined by putting for all , then (3) can be written as

Even when the partition is infinite, one may still hope to obtain a formula along the lines of (4).

Example 1 Let be the unit square, be two-dimensional Lebesgue measure, and be the horizontal line through . Then Fubini’s theorem gives (4) by taking to be Lebesgue measure on horizontal lines, and defining on , the set of horizontal lines in , in one of the two following (equivalent) ways:

1. given , let ;
2. identify with the interval on the -axis, and define as the image of one-dimensional Lebesgue measure on this interval.

Note that must satisfy the first of these no matter what the partition is, while the second is a convenient description of in this particular example.

A similar-looking example (and the one which is most relevant for our purposes) comes by letting be a rectangle and letting be the partition into local unstable leaves. To produce conditional measures , we need to use the fact that the partition is measurable. This means that there is a sequence of finite partitions that refines to in the sense that for -a.e. , we have

Lemma 4 Given any rectangle , the partition of into local unstable leaves is measurable.

Proof: Fix and let be a refining sequence of finite partitions of with the property that for all ; then let , and we are done.

Whenever is a measurable partition of a compact metric space , we can define the conditional measures as the limits of the conditional measures . Indeed, one can show (we omit the proofs) that for -a.e. , the limit exists for every continuous and defines a continuous linear functional ; the corresponding measures satisfy

for every , where once again we put .

The key result that we will need to describe properties of when is the partition of into local unstable leaves is that for -a.e. and every , we have

where is any sequence of finite partitions that refines to .

In order to establish the local product structure for that is claimed by Theorem 3, we will show that the measures vary in an absolutely continuous manner as varies within . That is, we consider for every the holonomy map defined by moving along local stable manifolds, so

Our goal is to use the Gibbs property (1) for to prove that for every rectangle and -a.e. , the conditional measures and satisfy

Once this is established, we can proceed as follows. Consider a rectangle with the decomposition into local unstable manifolds, let be such that (7) holds for -a.e. , and then identify with , as in the second characterization of in Example 1. Let be the measure on corresponding to under this identification, and let . Given , let , so that in particular we have

for every continuous . Then by (5), we have

By Definition 2, this shows that has local product structure with respect to . Thus in order to prove Theorem 3, it suffices to shows that Gibbs measures satisfy the absolute continuity property (7).

It is worth noting quickly that our use of the term “absolute continuity” here has a rather different meaning from another common concept, which is that of a measure with “absolutely continuous conditional measures on unstable manifolds”. This latter notion is essential for the definition of SRB measures (indeed, in the Anosov setting it is the definition), and involves comparing to volume measure on , instead of to the pushforwards of other conditional measures under holonomy.

3. Adapted partitions

In order to prove the absolute continuity property (7), we need to obtain estimates on . We start by getting estimates on from the Gibbs property (1), and then using these to get estimates on using (6).

We will need a family of partitions of that refines to the partition into points. Fix a reference point , and suppose we have chosen partitions of and of for . Then we can define a partition of by taking the direct product of these two partitions, using the foliations of by local stable and unstable leaves: that is, we put

In order to obtain information on using the Gibbs property (1), we need to put an extra condition on the partitions we use; we need to them to be adapted, meaning that each partition element both contains a Bowen ball and is contained within a larger Bowen ball. Most of the ideas here are fairly standard in thermodynamic formalism, but it is important for us to work separately on the stable and unstable manifolds, then combine the two, so we describe things explicitly. Fix . Given and , let

Similarly, given and , let

Given , we say that the partitions are -adapted if for every partition element there is such that

We make a similar definition for using . Note that we can produce an -adapted sequence of partitions as follows:

1. say that is -separated if for every there is such that ;
2. let be a maximal -separated set and observe that , while the sets are disjoint;
3. enumerate as , and build an -adapted partition by considering the sets

   

Lemma 5 Let be a measure satisfying the Gibbs property (1). Then there are and such that if and are -adapted partitions of and , then the product partition defined in (8) satisfies

for every .

Proof: First we show that the upper bound in (10) holds whenever is sufficiently small. Fix such that the Gibbs bound (1) holds for Bowen balls of radius , and such that is significantly smaller than the size of any local stable or unstable leaf. It suffices to show that for every we have for some , since then (1) gives the upper bound, possibly with a different constant; note that replacing with in the denominator changes the quantity by at most a constant factor, using the fact that is Hölder continuous together with some basic properties of Anosov maps. (See, for example, Section 2 of this previous post for a proof of this Bowen property.)

Given such that and lie on the same local stable leaf with , let

be the closest that any holonomy along local unstable leaves can bring and . Note that is positive and continuous in and ; by compactness there is such that for all as above. In particular, this means that if are the images under an (unstable) holonomy of some with , then we must have .

Choose similarly for stable holonomies, and fix . Fix , , , and . Then for every we have

These distances can be estimated by observing that and are the images of and under a holonomy map along local unstables, and similarly and are images of and under a holonomy map along local stables. Then our choice of shows that both quantities in the right-hand side of (11) are , which gives the inclusion we needed. This proves the upper bound in (10); the proof of the lower bound is similar.

4. A refining sequence of adapted partitions

Armed with the formula from Lemma 5, we may look at the characterization of conditional measures in (6) and try to prove that the absolute continuity bound (7) holds whenever both satisfy (6). There is one problem before we do this, though; the formula in (6) requires that the sequence of partitions be refining, and there is no a priori reason to expect that the adapted partitions produced by the simple argument before Lemma 5 refine each other. To get this additional property, we must do some more work. To the best of my knowledge, the arguments here are new.

4.1. Strategy

Start by letting be a maximal -separated set. We want to build a refining sequence of adapted partitions using the sets , where instead of using Bowen balls with radius and we will use Bowen balls with radius and ; this does not change anything essential about the previous section. We cannot immediately proceed as in the argument before Lemma 5, because if we are not careful when choosing the elements of the partition , their boundaries might cut the Bowen balls for and , spoiling our attempt to build an adapted partition that refines .

A first step will be to only build for some values of . More precisely, we fix such that , so that for every and , we have ; then writing

we have the following for every :

We will only build adapted partitions for those that are multiples of . We need to understand when the Bowen balls associated to points in the sets can overlap. We write .

Definition 6 An -path between and is a sequence with and such that for all . A subset is -connected if there is an -path between any two elements of .

Given , let . In the next section we will prove the following.

Proposition 7 If is -connected and , then

For now we show how to build a refining sequence of adapted partitions assuming (14) by modifying the construction in (9). The key is that we build our partitions so that for every -connected , the set is completely contained in a single element of .

Suppose we have built a partition with this property; we need to construct a partition that refines , still has this property, and also has the property that every partition element has some such that . To this end, let be an element of the partition , and enumerate the -connected components of as , where each contains some , while each is in fact a subset of .

It follows from (14) that for all . Given , we can observe that for some with . Moreover, the sets cover , so every must intersect some set , and hence be contained in some . Given , let . Then for each , let . Define sets by

It is not hard to verify that the sets form a partition of such that , and such that every -connected component of that has is completely contained in some . Repeating this procedure for the other elements of produces the desired .

4.2. Proof of the proposition

Now we must prove Proposition 7. Let be as in the previous section, so that in particular, every has that is a multiple of , and given any we have .

The following concept is essential for our proof.

Definition 8 Given an -path , a ford is a pair for all . Think of “fording a river'' to get from to by going through deeper levels of the -path; the alternative is that for some , in which case is a sort of “bridge'' between and .

Lemma 9 Suppose that is an -path without any fords, and that for all . Then .

Proof: By the definition of -path, for every there is a point . Thus

where the second inequality uses (13). Now we need to estimate how often different values of can appear. Let ; we claim that for every , we have

First note that for all , because otherwise would be a ford. Since the -path has no fords, every with must be separated by some with , and we conclude that for every , we have

For , this establishes (16) immediately. If (16) holds up to , then this gives

which proves (16) for all by induction. Combining (15) and (16), we have

which proves Lemma 9.

Now we show that the -separation condition rules out the existence of fords entirely.

Lemma 10 If is an -path, then it has no fords.

Proof: Fix an -path . Denote the set of fords by

our goal is to prove that is empty. Put a partial ordering on by writing

If is non-empty, then since it is finite it must contain some element that is minimal with respect to this partial ordering. In particular, the -path contains no fords, and so by Lemma 9 we have , contradicting the assumption that (since is -separated), and we conclude that must be empty, proving the lemma.

Now we prove Proposition 7. Let be -connected and fix . Given any , there is an -path such that and . By Lemma 10, this path has no fords, and so Lemma 9 gives . We conclude that , which proves Proposition 7.

5. Completion of the proof

Thanks to the previous section, we can let and be -adapted partitions of and such that refines whenever and are both multiples of . By Lemma 5, we have good lower and upper estimates on for all , where is the product partition defined in (8). By (6), there is with such that for every , we have

for every continuous . The next step towards proving Theorem 3 is the following result.

Proposition 11 There is a constant such that for any and every continuous , we have , where is the holonomy map along local unstables.

Proof: Start by fixing for each a set such that every element of contains exactly one point in . Then let for every .

Given a positive continuous function , there is such that if , then . Thus for all sufficiently large , we have whenever . For any such and any , we conclude that

where we write and where the last inequality uses Lemma 5. Similarly,

and we conclude from (6) that

Note that since is Hölder continuous, there are and such that for all . Thus given any and any , we have

Thus (18) gives

A similar set of computations for shows that

Since , we can rewrite the sums over as sums over ; for example,

where the inequality uses the fact that the estimate (19) also holds for forward ergodic averages of two points on the same local stable manifold. Using a similar estimate for the numerator in (21) gives

Together with (20), this gives

which completes the proof of Proposition 11.

To complete the proof of Theorem 3, we first observe that for every open set , there is a sequence of continuous functions that converge pointwise to the indicator function ; applying Proposition 11 to these functions and using the dominated convergence theorem gives

Then for every measurable , we have

This proves that and that the Radon–Nikodym derivative is -a.e. The lower bound follows since the argument is symmetric in and . This proves (7) and thus completes the proof of Theorem 3.

Alpha-beta shifts

Given , the interval map given by is naturally semi-conjugate to the -shift . The shift space admits a natural description in terms of the lexicographic order, and another in terms of a countable-state directed graph. These have been used to obtain a fairly complete description of the dynamics of the -transformation , including existence, uniqueness, and strong statistical properties of equilibrium states for Hölder potentials.

In this post I want to explain how one can give similar descriptions (both in terms of lexicographic order and in terms of a countable-state Markov structure) for the coding spaces associated to the interval map . Just as with the -shifts, these “– shifts” arise from piecewise expanding interval maps, and thus can be studied using the general machinery developed by Hofbauer, but I believe that they are worth studying a little more carefully as an intermediate case between the -shifts, where the fact that is a `safe symbol’ can be used to prove various results that are still open for – shifts, and the more general setting where the description of the shift space requires more bookkeeping and it can be harder to see what is going on.

1. Coding spaces for interval maps

We start by recalling the general notion of a coding space for a map of the interval. Say that a map is a piecewise expanding interval map if there is and a partition of into finitely many intervals such that is continuous on each and on the interior of each , with . Note that we do not care whether the intervals are closed, open, or half of each.

Let , and define a map by whenever . Let ; we say that is the coding space for the interval map relative to the partition . We can define a map by the condition that for all and is continuous. Then we have , so is a semiconjugacy between and .

Another way of interpreting this is the following: for each there is a unique map such that for all in the interior of . Note that is with . Given any set , write for convenience; note that if does not intersect . Given a finite word , let , where and is the length of . Roughly speaking, represents the set of points for which , , and so on. (This is not quite completely true because of problems at the endpoints of the intervals.)

The language of the shift is the set of all such that . Write for this collection of words; then , and we have for all . Let ; given , the interval has length , and since for every we have , we also have .

So far these considerations have been completely general, valid for any piecewise expanding interval map. Now we consider the specific transformations and , where and we assume without loss of generality that . These are piecewise expanding interval maps, where the natural partition to consider is the partition into maximal intervals of continuity. Thus for , we have , , and so on, with , where . For , we have , , and so on, with and . We write and for the coding spaces of these transformations relative to their natural partitions.

2. Description via lexicographic order

The full shift carries a natural lexicographic order inherited from the usual order on : given , let be minimal such that , and write if . Write if or . This is a total order on . The key observation for our purposes is that because and are order-preserving on each of their intervals of continuity, their coding maps are also order-preserving in the sense that if and only if . (Note that we must use non-strict inequalities because the coding maps are not 1-1.)

2.1. The -shifts

Fix and let . Let , where supremum is w.r.t. lexicographic order; the supremum exists because has the least upper bound property, and is in because the -shift is closed. Thus is the lexicographically maximal element of ; it would code the trajectory of the right endpoint of the unit interval if we replaced with the map , which agrees with everywhere except at the endpoints of the intervals of monotonicity. The points will play an important role in the following result.

Proposition 1 A sequence is in if and only if

Similarly, a word is in if and only if

Proof: Because is closed and -invariant, every satisfies (1), and it follows immediately that every satisfies (2).

To prove the converse direction, it suffices to show that every which satisfies (2) is contained in ; in other words, that for every such . Equivalently, we can work with the following object: consider for each the follower interval ; this can also be defined as the minimal interval with the property that . Note that whenever . Given and , observe that

Now we can give a non-recursive description of that completes the proof of Proposition 1. If satisfies (2), let

be the length of the longest prefix of that appears as a suffix of .

Lemma 2 If satisfies (2), then .

Proof: We go by induction in the length of , using (so is the empty word and ) as the base case. Suppose we have some satisfying (2), and that the lemma has been proved for all words with length . Write where and . Let , so that for some word . By the inductive hypothesis, we have and hence . Then by (3), we have

There are two cases to consider.

Case one: . In this case we have and , and .

Case two: . In this case we must have , otherwise we would have and hence , violating (2). Thus and hence by (3), we have . On the other hand, since , we see that for every we have

where the last inequality uses the fact that . We conclude that , which completes the proof of Lemma 2.

With Lemma 2 complete, Proposition 1 follows immediately since implies that .

A historical note: the transformation was introduced by Rényi in 1957 to study representations of real numbers in non-integer bases. The lexicographic characterization of -shifts in Proposition 1 was given by Parry in 1960 (see Theorem 3 of that paper) using different methods than the proof here. This proof is essentially the same as the one given by Hofbauer in 1979 (see Theorem 2 there); Hofbauer’s approach has the advantage of dealing with all piecewise monotonic interval maps, and the argument given here for the -shift is actually just a special case of his general result.

2.2. The – shifts

Instead of describing the full generality of Hofbauer’s result here, we describe how Proposition 1 adapts to the – shifts. This family of examples already reveals some of the challenges that arise when we go from -shifts to more general piecewise expanding transformations.

Fix and , then let , and let be the coding space for with respect to its natural partition. Let and be the lexicographically minimal and maximal elements of . Then every has the property that

and every has the property that

As with the -shifts, these conditions are both necessary and sufficient for membership in and . The key to proving this is the following analogue of Lemma 2, whose proof we omit since it is similar to the proof there (with just a little more bookkeeping).

Lemma 3 Given satisfying (6), let

Then , where and agree with on the interiors of the intervals , and are defined at the endpoints by the condition that is continuous from the right and is continuous from the left.

3. A directed graph on

The lexicographic description of in Proposition 1 is appealing, but for many purposes it is more useful to have a description of the -shift in terms of a countable-state directed graph, which goes back to work of Takahashi in 1973 and Hofbauer in 1978.

Start by considering the notion of follower set in a shift space, which is analogous to the follower intervals considered in the previous part. Given a shift space with language , consider for each the follower set

One could also consider the set of words such that , but the definition in (8) will be more convenient for our purposes. It was shown by Weiss in 1973 that a shift can be represented via a labeled directed graph on finitely many vertices if and only if it has finitely many follower sets; that is, if is finite. Such shifts are called sofic. Similarly, can be represented via a labeled irreducible directed graph on countably many vertices if and only if it has countably many follower sets, and such shifts are called coded; see Fiebig and Fiebig 1992 and 2002.

Every -shift is characterized by a sequence with the property that for all : as we saw in Proposition 1, we have

Moreover, it follows from Lemma 2 that for every , the corresponding follower set in is

where is defined in (4) to be the length of the longest prefix of that appears as a suffix or . Thus is completely determined by , which can take countably many values, and since -shifts are transitive this implies that they are coded. Thus they can be presented by a countable-state graph which supports a countable-state topological Markov chain (the Fischer cover).

Given any coded shift, the vertex set of the Fischer cover is the set of follower sets: for some . A labeled edge is a triple , where and ; the labeled edge set of the Fischer cover is ; that is, there is an edge from to whenever and , and this edge is labeled with the symbol . Note that there may be multiple edges between two vertices, each with a different label.

Say that a sequence labels a walk on the graph if there is a sequence of edges such that for every ,

1. is labeled with the symbol , and
2. the target vertex of is the source vertex of .

Then if labels a walk on this graph, there is a word such that for all , and hence . Conversely, every labels a walk on the graph, so we can describe both and in terms of walks on the graph.

For the -shift, since the follower set is completely determined by , we can identify the vertex set of the Fischer cover with , and observe that the two cases in Lemma 2 give the following two types of edges.

1. There is an edge from to whenever there are and such that , , and . This happens for every since we can take and , so the edge from to is labeled with the symbol .
2. There is an edge from to whenever there are and such that , , and . This happens whenever , and so whenever there are edges from to labeled with the symbols .

For example, when , the first part of the graph looks like this:

The vertex labeled 0 can be thought of as the base vertex; it corresponds to the follower set , so every sequence labels a walk starting at this vertex. The vertex labeled 1 corresponds to the follower set ; a sequence labels a walk starting at this vertex if and only if . For a more typical sort of example, let . Then corresponds to the walk which starts at vertex 0, goes out to vertex 2 before returning to vertex 0 (thanks to the fact that ), then goes out to vertex 4 before returning to vertex 0 (because ), then goes out to vertex 3 and stops. We see that and that

4. A directed graph on

In order to give an analogous description of via a countable-state Markov shift, we first observe that is determined lexicographically by two sequences with the property that

Given such an and , the corresponding shift space and language are given by

From Lemma 3, the follower sets are given by

where are given by (7) and represent the lengths of the longest prefixes of and , respectively, that appear as suffixes of . Thus the follower set of is completely determined by and , and so we can represent by a graph whose vertices lie in . As we will see momentarily, not every pair corresponds to a follower set, so we will not use all such vertices in building the graph, but thinking of the integer lattice still provides a useful way to visualize that situation.

It is useful to look at a concrete example. Let and . Then some of the follower sets, together with their corresponding values of , are as follows:

In the graph for the -shifts, remember that the outgoing edges from vertex came in two types: from to with label for each , and from to with label for , depending on whether or . (We switch to using for a generic element of now that is in play.)

For the – shifts, there are more possibilities for and :

1. , which happens if , and gives an edge from to labeled ;
2. and , which happens if , and gives an edge from to labeled ;
3. and , which happens if , and gives an edge from to labeled ;
4. and , which happens if , and gives an edge from to labeled .

For our concrete example with and , the first part of the graph is as follows. Note that our coordinate conventions follow standard matrix notation (down, then right) instead of the usual cartesian coordinates, and that we write the vertices as instead of , and so on, in order to save space.

Observe that there are two types of vertices: on the one hand there are those such as , , , , with exactly one outgoing edge, which goes down and to the right; on the other hand there are vertices which have at least two outgoing edges, each of which `resets’ in either the horizontal or vertical directions (or both).

Let us conclude by formulating an open problem. In a previous post, I gave a proof of the (well-known) result that if is a shift space with the specification property, then every Hölder continuous potential has the property that every equilibrium state for has positive entropy (“every Hölder potential is hyperbolic”). The -shifts do not have specification in general (Schmeling showed in 1997 that the set of such that has specification is a null set for Lebesgue measure), but it turns out that they still have the property that every Hölder potential is hyperbolic; Dan Thompson and I showed this in a 2013 paper (arXiv:1106.3575).

Our argument there (see Proposition 3.1 and Section 6.1) relies on the fact that 0 is a `safe symbol’ for the -shifts; if and is obtained from by replacing a single symbol with 0, then as well. This is no longer true for , and so our proof does not generalize. It would be interesting to know whether still has the property that every Hölder potential is hyperbolic; so far as I know this question is wide open. Buzzi conjectured in 2004 that this property holds not just for the – shifts, but for every coding space of a piecewise monotonic interval map. Attacking this problem for the – shifts seems like a reasonable first step towards resolving this conjecture.

Entropy bounds for equilibrium states

[Update 6/15/17: The original version of this post had a small error in it, which has been corrected in the present version; the definition of in the proof of the main theorem needed to be modified so that each is a multiple of .  Thanks to Leonard Carapezza for pointing this out to me.]

Let be a compact metric space and a homeomorphism. Recall that an equilibrium state for a continuous potential function is an -invariant Borel probability measure on maximizing the quantity over all invariant probabilities; the topological pressure is the value of this maximum.

A classical result on existence and uniqueness of equilibrium states is due to Bowen, who proved that if is expansive and has specification, and has a bounded distortion property (the `Bowen property’), then there is a unique equilibrium state . In particular, this applies when is Anosov and is Hölder.

It seems to be well-known among experts that under Bowen’s hypotheses, must have positive entropy (equivalently, ), but I do not know of an explicit reference. In this post I provide a proof of this fact, which also gives reasonably concrete bounds on the entropy of ; equivalently, a bound on the size of the gap .

1. Definitions and result

First, let’s recall the definitions in the form that I will need them. Given , , and , the Bowen ball around of order and radius is the set

The map has specification if for every there is such that for every and , there is such that

and in general

for every . We refer to as the “gluing time”; one could also consider a weaker property where the gluing times are allowed to vary but must be bounded above by ; this makes the estimates below more complicated, so for simplicity we will stick with the stronger version.

A function has the Bowen property at scale with distortion constant if is such that

where . We write

where is the collection of -separated subsets of (those sets for which whenever , ). The topological pressure is , where

Theorem 1 Let be a compact metric space with diameter , a homeomorphism with specification at scale with gap size , and a potential with the Bowen property at scale with distortion constant . Let

where . Then we have

 

In particular, if is an equilibrium state for , then we have .

2. Consequence for Anosov diffeomorphisms

Before proving the theorem we point out a useful corollary. If is a compact manifold and is a topologically mixing Anosov diffeomorphism, then has specification at every scale (similar results apply in the Axiom A case). Moreover, every Hölder continuous potential has the Bowen property, and thus Theorem 1 applies.

For an Anosov diffeo, the constants and in (1) can be controlled by the following factors (here we fix a small ):

1. the rate of expansion and contraction along the stable and unstable directions, given in terms of such that for all and , and similarly for and ;
2. how quickly unstable manifolds become dense, in other words, the value of such that is -dense for every choice of ;
3. the angle between stable and unstable directions, which controls the local product structure, in particular via a constant such that implies that intersects in a unique point , and the leafwise distances from to are at most ;
4. the Hölder exponent () and constant () for the potential .

For the specification property for an Anosov diffeo, is determined by the condition that , so that small pieces of unstable manifold expand to become -dense within iterates; thus we have

For the Bowen property, one compares and by comparing each to , where is the (Smale bracket) intersection point coming from the local product structure. Standard estimates give , so the Hölder property gives

A similar estimate for gives

Thus Theorem 1 has the following consequence for Anosov diffeomorphisms.

Corollary 2 Let be a topologically mixing Anosov diffeomorphism on and the quantities above. Let

Given a -Hölder potential , consider the quantity

Then we have

so that in particular, if is an equilibrium state for , then

Finally, note that since shifting the value of by a constant does not change its equilibrium states, we can assume without loss of generality that and write the following consequence of the above, which is somewhat simpler in appearance.

Corollary 3 Let be a compact manifold and a topologically mixing Anosov diffeomorphism. For every there are constants and such that for every -Hölder potential , we have

so that as before, if is an equilibrium state for , we have

This corollary gives a precise bound on how the entropy of a family of equilibrium states can decay as the Hölder semi-norms of the corresponding potentials become large. To put it another way, given any threshold , this gives an estimate on how large must be before can have an equilibrium state with entropy below .

3. Proof of the theorem

We spend the rest of the post proving Theorem 1. Fix and consider for each the orbit segment . Fix . Let , and let

Write and . The idea is that for each , we will use the specification property to construct a point whose orbit shadows the orbit of from time to time , except for the times , at which it deviates briefly; thus the points will be -separated on the one hand, and on the other hand will have ergodic averages close to that of .

First we estimate from below; this requires a lower bound on . Integrating over and gives

and thus we have

where . This function is increasing on , so

 

Given , let be any point with (using the assumption on the diameter of ). Now for every , the specification property guarantees the existence of a point with the property that

and so on, so that in general for any we have

 

Write ; then the first inclusion in (3), together with the Bowen property, gives

Now observe that for any we have

We conclude that

 

Consider the set . The second inclusion in (3) guarantees that this set is -separated; indeed, given any , we can take to be minimal such that , let , and then observe that and ; since this guarantees that .

Using this fact and the bounds in (4) and (2), we conclude that

Taking logs, dividing by , and sending gives

Given any ergodic , we can take a generic point for and conclude that the lim sup in the above expression is equal to . Thus to bound the difference , we want to choose the value of that maximizes , where .

A straightforward differentiation and some routine algebra shows that occurs when , at which point we have , proving Theorem 1.

Exponential decay of correlations

Let be a probability space and a measure-preserving transformation. Let be Banach spaces of measurable functions on , and the corresponding norms. Given and , the corresponding th correlation is

We say that has exponential decay of correlations with respect to observables in if for any , we have

 

Equivalently, for every and , there are and such that

 

One sometimes sees the statement of exponential decay given in the following form, which is formally stronger than (2): there are constants and , independent of , such that

 

In fact, using the Baire category theorem, one can prove that (2) implies (3) under a mild condition on the Banach spaces ; this is the goal of this post, to show that can be chosen uniformly over all , and that can be chosen to have the form . This seems like the sort of thing which is likely known to experts, but I am not aware of the reference in the literature. (I would be happy to learn a reference!)

Proposition 1 Let be a probability measure-preserving transformation and Banach spaces of measurable functions on with the following properties:

• given any , we have and ;
• the inclusions are continuous;
• for every , the map given by is continuous, and similarly for the map when is fixed;
• the map given by is bounded w.r.t. .

Under these assumptions, if has exponential decay of correlations w.r.t. observables in in the sense of (2), then it also satisfies (3).

To prove the proposition, start by fixing and , and consider the function given by

Notice that for each , the correlation function is bilinear in , and thus for every and , we have

 

Consider the following subsets of :

It follows from (2) that

 

Moreover, the sets are nested (smaller gives a bigger set, larger gives a bigger set) and so it suffices to take the union over rational values of , meaning that we can treat (5) as a countable union. In particular, by the Baire category theorem there are such that the closure of has non-empty interior. The next step is to show that

• is closed, so it itself has non-empty interior;
• in fact, contains a neighbourhood of the origin.

For the first of these, observe that by the assumptions we placed on the Banach spaces , there is a constant such that

for every . In particular,

Given a sequence such that w.r.t. as , it follows that

and we conclude that , so this set is closed. In particular, there is and such that if , then . By the same token , and now the sublinearity property (4) gives

and so . This shows that contains a neighbourhood of 0, and writing , we see that for every we have , and so

Thus we conclude that

To complete the proof of the proposition, it suffices to apply the same argument once more. Writing

we see from (2) that , and so once again there are such that the closure of has non-empty interior. Given a sequence with , we have for all and all , and so , demonstrating that this set is closed.

Thus there are and such that implies . In particular this gives , and so for every and every we have

But then for every we can consider , which has , and so

which proves (3) and the proposition.

Lebesgue probability spaces, part II

This is a continuation of the previous post on the classification of complete probability spaces. Last time we set up the basic terminology and notation, and saw that for a complete probability space , the -algebra is countably generated mod 0 (which we denoted (CG0)) if and only if the pseudo-metric makes into a separable metric space. We also considered as a measured abstract -algebra with non non-trivial null sets; this is the point of view we will continue with in this post.

Our goal is to present the result from [HvN] and [Ro] (see references from last time) that classifies such measured abstract -algebras up to -algebra isomorphism. To this end, let be a separable measured abstract -algebra with no non-trivial null sets; let be the maximal element of , and suppose that . Note that is the minimal element of , which would correspond to if were a collection of subsets of some ambient space. In the abstract setting, we will write for this minimal element.

An element is an atom if it has no proper non-trivial subelement; that is, if implies or . By the assumption that has no non-trivial null sets, we have for every atom. Note that for any two atoms we have , and so .

Let be the set of atoms; then we have , so is (at most) countable. Let

then is non-atomic; it contains no atoms. Thus can be decomposed as a non-atomic part together with a countable collection of atoms. In particular, to classify we may assume without loss of generality that is non-atomic.

Consider the unit interval with Lebesgue measure; let denote the set of equivalence classes (mod 0) of measurable subsets of , and denote Lebesgue measure, so is a measured abstract -algebra. Moreover, is separable, non-atomic, has no non-trivial null sets, and has total weight 1.

Theorem 1 Let be a separable non-atomic measured abstract -algebra with total weight 1 and no non-trivial null sets. Then is isomorphic to .

The meaning of “isomorphism” here is that there is a bijection that preserves the Boolean algebra structure and carries to . That is: iff ; ; ; ; ; and finally, . We are most used to obtaining morphisms between -algebras as a byproduct of having a measurable map between the ambient sets; that is, given measurable spaces and , a measurable map gives a morphism . However, in the abstract setting we currently work in, there is no ambient space, so we cannot yet interpret this way. Eventually we will go from and ( back to the measure spaces that induced them, and then we will give conditions for to be induced by a function between those spaces, but for now we stick with the abstract viewpoint.

We sketch a proof of Theorem 1 that roughly follows [HvN, Theorem 1]; see also [Ro, §1.3]. Given a measurable set , we write for the equivalence class of ; in particular, we write for the equivalence class of the interval .

Observe that if is dense, then is generated by ; thus we can describe by finding such that

• ,
• whenever ,
• generates ,

and then putting . This is where separability comes in. Let be a countable collection that generates . Put , so for . Now what about ? We can use to define two more of the sets by noting that

and so we may reasonably set for .

To extend this further it is helpful to rephrase the last step. Consider

and observe that (1) can be rewritten as

Writing for the lexicographic order on , we observe that each of these three is of the form for some .

Each above is determined as follows: tells us whether to use or , and tells us whether to use or . This generalises very naturally to : given , let

Now put ; this gives elements (the last one is , and was omitted from our earlier bookkeeping). These have the property that whenever . Moreover, writing , the collection generates because does. Let ; now we have all the pieces of the construction that we asked for earlier.

Putting it all together, let . The following are now relatively straightforward exercises.

• is dense in since is non-atomic.
• For each there is with ; if such that this holds, then either or (or vice versa). For this step we actually need to choose a little more carefully to guarantee that each is non-trivial.
• The previous step guarantees that is a bijection between and .
• Since preserves the order on the generating collections, it preserves the -algebra structure as well.
• Since carries to on the generating collections, it carries to on the whole -algebra.

Thus we have produced a -algebra isomorphism , proving Theorem 1. Next time we will discuss conditions under which this can be extended to an isomorphism of the measure spaces themselves, and not just their abstract measured -algebras.

Lebesgue probability spaces, part I

In various areas of mathematics, classification theorems give a more or less complete understanding of what kinds of behaviour are possible. For example, in linear algebra we learn that up to isomorphism, is the only real vector space with dimension , and every linear operator on a finite-dimensional vector space can be put into Jordan normal form via a change of coordinates; this means that many questions in linear algebra can be answered by understanding properties of Jordan normal form. A similar classification result is available in measure theory, but the preliminaries are a little more involved. In this and the next post I will describe the classification result for complete probability spaces, which gives conditions under which such a space is equivalent to the unit interval with Lebesgue measure.

The main original references for these results are a 1942 paper by Halmos and von Neumann [“Operator methods in classical mechanics. II”, Ann. of Math. (2), 43 (1942), 332–350, and a 1949 paper by Rokhlin [“On the fundamental ideas of measure theory”, Mat. Sbornik N.S. 25(67) (1949). 107–150, English translation in Amer. Math. Soc. Translation 1952 (1952). no. 71, 55 pp.]. I will refer to these as [HvN] and [Ro], respectively.

1. Equivalence of measure spaces

First we must establish exactly which class of measure spaces we work with, and under what conditions two measure spaces will be thought of as equivalent. Let be the unit interval and the Borel and Lebesgue -algebras, respectively; let be Lebesgue measure (on either of these). To avoid having to distinguish between and , let us agree to only work with complete measure spaces; this is no great loss, since given an arbitrary metric space we can pass to its completion .

The most obvious notion of isomorphism is that two complete measure spaces and are isomorphic if there is a bijection such that are measurable and ; that is, given we have if and only if , and in this case .

In the end we want to loosen this definition a little bit. For example, consider the space of all infinite binary sequences, equipped with the Borel -algebra associated to the product topology (or if you prefer, the metric ). Let be the -Bernoulli measure on ; that is, for each the cylinder gets weight . Then there is a natural correspondence between the completion and given by

By looking at dyadic intervals one can readily verify that ; however, is not a bijection because for every we have .

The points at which is non-injective form a -null set (since there are only countably many of them), so from the point of view of measure theory, it is natural to disregard them. This motivates the following definition.

Definition 1 Two measure spaces and are isomorphic mod 0 if there are measurable sets and such that , together with a bijection such that are measurable and .

From now on we will be interested in the question of classifying complete measure spaces up to isomorphism mod 0. The example above suggests that is a reasonable candidate for a `canonical’ complete measure space that many others are equivalent to, and we will see that this is indeed the case.

Notice that the total measure is clearly an invariant of isomorphism mod 0, and hence we restrict our attention to probability spaces, for which .

2. Separability, etc.

Let be a probability space. We describe several related conditions that all give senses in which can be understood via countable objects.

The -algebra carries a natural pseudo-metric given by . Write if ; this is an equivalence relation on , and we write for the space of equivalence classes. The function induces a metric on in the natural way, and we say that is separable if the metric space is separable; that is, if it has a countable dense subset.

Another countability condition is this: call “countably generated” if there is a countable subset such that is the smallest -algebra containing . We write (CG) for this property; for example, the Borel -algebra in satisfies (CG) because we can take to be the set of all intervals with rational endpoints. (In [HvN], such an is called “strictly separable”, but we avoid the word “separable” as we have already used it in connection with the metric space .)

In and of itself, (CG) is not quite the right sort of property for our current discussion, because it does not hold when we pass to the completion; the Lebesgue -algebra is not countably generated (one can prove this using cardinality estimates). Let us say that satisfies property (CG0) (for “countably generated mod 0”) if there is a countably generated -algebra with the property that for every , there is with . In other words, we have . Note that is countably generated mod 0 by taking . (In [HvN], such an is called “separable”; the same property is used in §2.1 of [Ro] with the label , rendered in a font that I will not attempt to duplicate here.)

In fact, the approximation of by satisfies an extra condition. Let us write (CG0+) for the following condition on : there is a countably generated such that for every , there is with and . This is satisfied for and . (In [HvN], such an is called “properly separable”; the same property is used in §2.1 of [Ro] with the label .)

The four properties introduced above are related as follows.

The first two implications are immediate, and their converses fail in general:

• The Lebesgue -algebra satisfies (CG0+) but not (CG).
• Let . Then satisfies (CG0) but not (CG0+).

Now we prove that (CG0) and separability are equivalent. First note that if is a countable subset, then the algebra generated by is also countable; in particular, is separable if and only if there is a countable algebra that is dense with respect to , and similarly in the definition of (CG0) the generating set can be taken to be an algebra. To show equivalence of (CG0) and separability it suffices to show that given an algebra and , we have

First we prove by proving that is a -algebra, and hence contains ; this will show that (CG0) implies separability.

• Closure under : if then there are such that . Since and (since it is an algebra), this gives .
• Closure under : given , let . To show that , note that given any , there are such that . Let and ; note that

  

  Moreover by continuity from below we have , so , and thus for sufficiently large we have . This holds for all , so .

Now we prove , thus proving that is “large enough” that separability implies (CG0). Given any , there are such that . Let We get

and similarly, , which gives

Then , which completes the proof of .

The first half of the argument above (the direction) appears in this MathOverflow answer to a question discussing the relationship between different notions of separability, which ultimately inspired this post. That answer (by Joel David Hamkins) also suggests one further notion of “countably generated”, distinct from all of the above; say that satisfies (CCG) (for “completion of countably generated”) if there is a countably generated -algebra such that , where is the completion of with respect to the measure . One quickly sees that

Both reverse implications fail; the Lebesgue -algebra satisfies (CCG) but not , and an example satisfying separability (and hence (CG0)) but not (CCG) was given in that same MathOverflow answer (the example involves ordinal numbers and something called the “club filter”, which I will not go into here).

3. Abstract -algebras

It is worth looking at some of the previous arguments through a different lens, that will also appear next time when we discuss the classification problem.

Recall the space of equivalence classes from earlier, where means that . Although elements of are not subsets of , we can still speak of the “union” of two such elements by choosing representatives from the respective equivalence classes; that is, given , we choose representatives and (so ), and consider the “union” of and to be the equivalence class of ; write this as . One can easily check that this is well-defined; if and , then .

This shows that induces a binary operation on the space ; similarly, induces a binary operation , complementation induces an operation , and set inclusion induces a partial order . These give the structure of a Boolean algebra; say that is an abstract Boolean algebra if it has a partial order , binary operations , , and a unary operation , satisfying the same rules as inclusion, union, intersection, and complementation:

1. is the join of and (the minimal element such that ), and is the meet of and (the maximal element such that );
2. the distributive laws and hold;
3. there is a maximal element whose complement is the minimal element;
4. and .

For the form of this list I have followed this blog post by Terry Tao, which gives a good in-depth discussion of some other issues relating to concrete and abstract Boolean algebras and -algebras.

Exercise 1 Using the four axioms above, prove the following properties:

• is the unique element satisfying (4) — that is, if and , then ;
• ;
• de Morgan’s laws: and .

If you get stuck, see Chapter IV, Lemma 1.2 in A Course in Universal Algebra by Burris and Sankappanavar.

In fact inherits just a little bit more, since (and hence ) can be iterated countably many times. We add this as a fifth axiom, and say that an abstract Boolean algebra is an abstract -algebra if in addition to (1)–(4) it satisfies

\setcounter{enumi}{4}
1. any countable family has a least upper bound and a greatest lower bound .

A measured abstract -algebra is a pair , where is an abstract -algebra and is a function satisfying the usual properties: and whenever for all . (Note that is playing the role of , but we avoid the latter notation to remind ourselves that elements of do not need to be represented as subsets of some ambient space.)

The operations induce a binary operator on by

which is the abstract analogue of set difference, and so a measured abstract -algebra carries a pseudo-metric defined by

If has the property that for all , then this becomes a genuine metric.

In particular, if is a measure space and is the space of equivalence classes modulo (equivalence mod 0), then induces a function , which we continue to denote by , such that is a measured abstract -algebra; this has the property that for all non-trivial , and so it defines a metric as above.

Given an abstract -algebra and a subset , the algebra (-algebra) generated by is the smallest algebra (-algebra) in that contains . Now we can interpret the equivalence (1) from the previous section (which drove the correspondence between (CG0) and separability) in terms of the measured abstract -algebra .

Proposition 2 Let be a measured abstract -algebra with no non-trivial null sets. Then for any algebra , we have ; that is, the -closure of is equal to the -algebra generated by .

Next time we will see how separability (or equivalently, (CG0)) can be used to give a classification result for abstract measured -algebras, which at first requires us to take the abstract point of view introduced in this section. Finally, we will see what is needed to go from there to a similar result for probability spaces.