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 1The best example to keep in mind through this talk is atopological Markov shift: fix and put ; then fix atransition matrixwith entries in , and writeDefine 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 ismixingorprimitiveif 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 1hasspecificationif there is such that for all there is such that .

Exercise 1Prove 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 2ProveFekete’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 3Prove 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 4Show 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 5Show 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 4is aGibbs measureif 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 5If is an ergodic Gibbs measure for , then and is the unique MME.

Theorem 6If 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 7Given 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 1It 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 6Show 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

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

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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.

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

Moreover, since integration is linear in , we have

Proposition 2if 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 1For 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:

- for some ;
- for all ;
- 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.

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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 CoudÃ¨ne, 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 HÃ¶lder 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 HÃ¶lder 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.

- There are and such that for all , , and , we have ; a similar contraction bound holds going backwards in time when .
- There is such that if for all , then ; similarly for with .
- 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 1For 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 2A measure haslocal product structurewith respect to if for every and rectangle there are measures on such that .

Theorem 3Let be a Anosov diffeomorphism, a HÃ¶lder 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 HÃ¶lder 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 CoudÃ¨ne’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 1Let 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:

- given , let ;
- 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 4Given 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:

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

Lemma 5Let 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

*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 HÃ¶lder 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 6An-pathbetween and is a sequence with and such that for all . A subset is-connectedif there is an -path between any two elements of .

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

Proposition 7If 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 8Given an -path , afordis 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 9Suppose 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.

*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 11There 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 HÃ¶lder 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.

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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 1A sequence 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 2If 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 3Given 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 ,

- is labeled with the symbol , and
- 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.

- 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 .
- 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 :

- , which happens if , and gives an edge from to labeled ;
- and , which happens if , and gives an edge from to labeled ;
- and , which happens if , and gives an edge from to labeled ;
- 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.

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*[ Update 6/15/17: The original version of this post had a small error in it, which has been corrected in the present version; *

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 HÃ¶lder.

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 1Let 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

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 HÃ¶lder 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 ):

- 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 ;
- how quickly unstable manifolds become dense, in other words, the value of such that is -dense for every choice of ;
- 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 ;
- the HÃ¶lder 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 HÃ¶lder property gives

A similar estimate for gives

Thus Theorem 1 has the following consequence for Anosov diffeomorphisms.

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

Given a -HÃ¶lder 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 3Let be a compact manifold and a topologically mixing Anosov diffeomorphism. For every there are constants and such that for every -HÃ¶lder 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 HÃ¶lder 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

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.

]]>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 1Let 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.

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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 1Let 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.

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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 1Two measure spaces and areisomorphic mod 0if 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:

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

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The variational principle for topological entropy says that if is a compact metric space and is a continuous map, then , where is the topological entropy, and the supremum is taken over all -invariant Borel probability measures. A measure achieving this supremum is called a *measure of maximal entropy* (MME for short), and it is interesting to understand when a system has a unique MME.

Let’s look at this question in the setting of symbolic dynamics. Let be a finite set, which we call an *alphabet*, and let be the set of all infinite sequences of symbols in . This is a compact metric space in a standard way, and the shift map defined by is continuous. We consider a compact -invariant set and ask whether or not has a unique MME.

When is a mixing subshift of finite type (SFT), this was answered in the 1960’s by Parry; there is a unique MME, and it can be obtained by considering the transition matrix for and using some tools from linear algebra. A different proof was given by Bowen in 1974 using the *specification* property; this holds for all mixing SFTs, but also for a more general class of shift spaces.

The purpose of this note is to describe a variant of Bowen’s proof; roughly speaking, we follow Bowen’s proof for the first half of the argument, then give a shorter version of the second half of the argument, which follows comments made in conversation with Dan Thompson and Mike Hochman.

**1. The strategy **

The *language* of the shift space is the set of all finite words that appear in some element of . That is, given we write , where . Then we write . Write for the length of .

In the setting of symbolic dynamics, the topological transitivity property takes the following form: is transitive if and only if for every there is such that . Transitivity by itself gives no control over the length of . We say that shift space has *specification* if there is such that for every there is with such that ; thus specification can be thought of as a uniform transitivity property.

Let be the topological entropy of . Bowen’s proof of uniqueness has the following structure:

- Show that there is such that for every .
- Prove that for every there is such that if is an MME and is a collection of words with , then we have . This can be thought of as a uniform version of the Katok entropy formula.
- Follow the proof of the variational principle to explicitly construct an MME ; then use the specification property and the counting estimates on to show that has the Gibbs property and is ergodic.
- Show that if is another ergodic MME, then the uniform Katok entropy formula for and the Gibbs property for cannot hold simultaneously; this contradiction proves uniqueness.

The proof we give here follows the above structure exactly for steps 1 and 2. Instead of steps 3 and 4, though, we give the following argument; if are two distinct ergodic MMEs, then we can use the uniform Katok entropy formula for together with the specification property to create more entropy in the system, a contradiction. In the next section we make all of this precise.

The advantage of this proof is that it allows us to replace step 3 of Bowen’s proof with a different argument that may be easier and less technical (depending on one’s taste). The disadvantage is that it does not include a proof of the Gibbs property for , which is useful to know about in various settings.

**2. The proof **

** 2.1. Counting estimates **

Write for the set of all words of the form , where and . Then it is clear that , so . In particular, is subadditive, so by Fekete’s lemma , and we get for every .

The upper bound requires the specification property. Define a map by sending to , where are provided by specification. This map is 1-1 so . Taking logs gives

and sending takes the left-hand side to , so ; this gives the counting bounds we claimed earlier, with .

The gluing construction in the previous paragraph will be used later on when we need to derive a contradiction by producing extra entropy.

** 2.2. Uniform Katok formula **

Now suppose is any MME, and given write . Similarly write . Applying Fekete’s lemma to the subadditive sequence , we get .

Given with , we have

Solving for gives

so taking with gives , verifying the uniform Katok formula.

Note that the argument here does not rely directly on the specification property; it only uses the fact that is an MME and that we have the uniform upper bound . In fact, if one is a little more careful with the computations it is possible to show that we can take , where . This has the pleasing property that as , so the lower bound for converges to the lower bound for .

** 2.3. Producing more entropy **

Now suppose are two distinct ergodic MMEs. Then so there are disjoint sets with and . Fix and let and be compact sets such that and . Then the distance between is positive, so there is such that for every , no -cylinder intersects both . In particular, putting , and similarly for with , we have

- for every , and
- and , so by the uniform Katok formula we have , where .

Up to now all of the arguments that we have made appear in Bowen’s proof; the approximation argument just given appears in step 4 of his proof, where one uses the Gibbs property of the constructed MME to derive a contradiction with the uniform Katok formula. It is at this point that our argument diverges from Bowen’s, since we have not proved a Gibbs property. Instead, we apply the specification property to the collections of words to get a lower bound on that grows more quickly than , a contradiction.

Before giving the correct argument, let me describe three incorrect arguments that illustrate the main ideas and also show why certain technical points arise in the final argument; in particular this will describe the process of arriving at the proof, which I think is worthwhile.

**First wrong argument**

Here is a natural way to proceed. With as above, consider for each and each the set of all words

where if and if , and are the gluing words provided by specification. Note that each choice of produces at least words in . Moreover, the collections of words produced by different choices of are disjoint, because and are disjoint. We conclude that

so

If then this would be enough to show that , a contradiction. Unfortunately since this argument does not work if , so we must try again…

**Second wrong argument**

Looking at the above attempted proof, one may describe the problem as follows: each of the words makes us lose a factor of from our estimate on . If , then instead of letting and range over and then gluing them, we could replace the words with the words . In particular, this would replace the estimate with the estimate .

This suggests that given , we should only keep track of the set for which , since if we can avoid losing the factor of by avoiding the gluing process.

So, let’s try it. Given , let (with and ), and consider the map

given by specification (whether the product ends with or depends on the parity of ).

Let be the image of , and note that

If the collections , were disjoint for different choices of , then we could fix and sum (1) over all with to get

where we use Sitrling’s approximation for the last inequality. Taking logs and dividing by gives

For sufficiently small this is , which gives , a contradiction.

The problem with this argument is that the collections need not be disjoint for different choices of ; this is because may have for some value of , so that we cannot necessarily recover uniquely from knowing .

**Third wrong argument**

Let’s try to address the issue just raised, that we cannot recover uniquely from because may have subwords in . We address this by only using words where there are `not many’ such subwords. More precisely, given for , let

be the set of `bad’ times, and similarly with the roles of and reversed. Let , and observe that since , invariance of gives

for every . We conclude that

Let , and note that

so

We conclude that

so taking , the uniform Katok estimate gives . A similar definition and argument gives .

Now we repeat the argument from the previous section (the second wrong argument) using in place of . Given , let be the image of the map with the corresponding restricted domain, and note that the estimate (1) still holds (with the new value of ).

The final piece of the puzzle is to take and estimate how many other collections can contain it; that is, how many possibilities there are for once we know . We would like to do the following: write and then argue that must be contained in the union of the sets of times corresponding to the . The problem is that this only works when there is a disagreement between which of the sets the maps are trying to use, and so I cannot quite make this give us the bounds we want.

**The correct argument**

To fix this final issue we change the construction slightly; instead of letting mark the times where we transition between , we do a construction where at each we transition from to and then immediately back to . Then will impose strong conditions on the set .

Let’s make everything precise and give the complete argument. As in the very beginning of this section we fix and let be disjoint compact sets with , where are distinct ergodic MMEs. Let be such that for every , no -cylinder intersects both .

Let , and similarly for . We repeat verbatim part of the argument from the last section; given for , let

be the set of `bad’ times. Let , and observe that since , invariance of gives

for every . We conclude that

Let , and note that

so

We conclude that

so taking , the uniform Katok estimate gives .

Now given and , let for (putting and ) and define a map

by the specification property, so that for and we have

where have length and are provided by specification. Let be the image of and note that

Given and , let denote the set of such that . Writing , note that implies that for each , either or there are consecutive elements of such that , and in this latter case we have that , so . We conclude that

By our choice of , for each the set on the right has at most elements. In particular, we have

This bounds the number of that can contain a given , and since there are distinct choices of with , the bound in (2) gives

Taking logs gives

Dividing by and sending gives

Putting this gives

Thus it suffices to make the appropriate choice of at the beginning of the proof. More precisely, let be as in the uniform Katok lemma, and let be small enough that . Then and so the estimate above gives

which contradicts our original assumption that was the topological entropy. This contradiction shows that there is a unique measure of maximal entropy.

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*alphabet*, and let be an infinite sequence of letters from . It is natural to ask how complex the sequence is: for example, if the alphabet is , then we expect a typical sequence produced by flipping a coin to be in some sense more complex than the sequence .

One important way of making this notion precise is the *entropy* of the shift space generated by , a notion coming from symbolic dynamics. Let be the number of words of length (that is, elements of ) that appear as subwords of . Clearly we have . Roughly speaking, the entropy of is the exponential growth rate of . More precisely, we write

Of course, in practice it is often the case that one does not have an infinite sequence, but merely a very long one. For example, it has been suggested that entropy (and a related quantity, the topological pressure) can play a role in the analysis of DNA sequences; see [D. Koslicki, “Topological entropy of DNA sequences”, Bioinformatics 27(8), 2011, p. 1061–1067] and [D. Koslicki and D.J. Thompson, “Coding sequence density estimation via topological pressure”, J. Math. Biol. 70, 2015, p. 45–69]. In this case we have and are dealing with sequences whose length is large, but finite.

Given a sequence with length , one can try to get some reasonable `entropy-like’ quantity by fixing some and putting . But what should we take to be? If we take to be too small we will get an overestimate (with we will probably just find out that contains every letter of ), but if we take too small we get an underestimate (with we have so ).

The convention proposed by Koslicki in the first paper above is to let be the largest number such that there is some word of length that contains every word of length . If this actually happens, then achieves its maximum value ; if some words do not appear, then .

What is the relationship between and that guarantees existence of a word of length containing every word of length ? Let and note that there are words of length ; if contains every such word then we must have , since the length- subwords of are precisely , , \dots, , so we must have .

The converse implication is a little harder, though. Given , let . Is it necessarily true that there is a word that contains every subword of length ? After all, once is determined, there are not many possibilities for the word ; can we navigate these restrictions successfully?

It is useful to rephrase the problem in the language of graph theory (what follows can be found in the proof of Lemma 1 in Koslicki’s paper). Let be the directed graph defined as follows:

- the vertex set is , so each vertex corresponds to a word of length ;
- there is an edge from to if and only if , that is, if .

The graph is the -dimensional De Bruijn graph of symbols. Recall that a Hamiltonian path in a graph is a path that visits each vertex exactly once. Thus the question above, regarding existence of a word in that contains every word in , where , is equivalent to asking for the existence of a Hamiltonian path in the De Bruijn graph.

There is a correspondence between and ; vertices in correspond to edges in (since both are labeled by elements of ). Thus a Hamiltonian path in corresponds to an Eulerian path in ; that is, a path that visits every edge exactly once.

This correspondence is very useful, since in general the problem of determining whether a Hamiltonian path exists is hard (NP-complete), while it is easy to check existence of an Eulerian path in a directed graph: a sufficient condition is that every vertex have in-degree equal to its out-degree (and a slight weakening of this condition is both necessary and sufficient). This is the case for De Bruijn graphs, where every vertex has edges coming in and going out. Thus has an Eulerian path, which corresponds to a Hamiltonian path for . This answers the original question, demonstrating that for every , there is a word of length such that contains every word of length as a subword.

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