In the last post we saw that if is the space of invariant probability measures for the full shift, then the collection of ergodic measures (which are the extreme points of the simplex ) has two remarkable properties:
- is dense in ;
- is arc-connected.
Moreover, it turned out that the first of these implies the second.
Arc-connectedness of gives a concrete sense in which the full shift is measure-theoretically indecomposable. It is natural to ask what the relationship is between the topological and measure-theoretic notions of irreducibility: does every topologically transitive system have the above properties? Do the above properties imply topological transitivity?
An extreme example of a non-transitive system is the disjoint union of two shifts. Here remains connected (as always), but the set becomes disconnected, and so we see that at least in certain extreme cases, topological and measure-theoretic irreducibility seem related. What happens in the middle?
We can take a step forward by recalling the paper of Sigmund quoted in the last post, which showed density of in for the full shift. In fact, Sigmund showed rather more — he showed that this density holds for every topological dynamical system with the specification property. To introduce this property, we first formulate topological transitivity in a mildly non-standard way.
Definition 1 Let be a compact metric space and a continuous map. Given , , and , the Bowen ball of order and radius centred at is
Thus a Bowen ball contains all points that shadow the orbit of for a given number of iterates with a given precision.
Proof: If is transitive, then there exists such that is dense. In particular, since Bowen balls are open, for each there exist arbitrarily large values of such that . Thus lies in the given intersection for an appropriate choice of .
On the other hand, if there always exist such that the intersection in~(1) is non-empty, then for any two open sets and we can choose such that and . Taking shows that for some , which gives transitivity.
The proposition shows that topological transitivity can be thought of as the ability to shadow an arbitrary collection of orbit segments with a single orbit, provided one is willing to wait a while between one orbit segment and the next. To put it another way, arbitrary orbit segments can be concatenated into a single orbit, up to a small error (of size ) and the insertion of a gap (of length ) between consecutive segments, during which time we have no control over the orbit.
Last time we saw that for the full shift, density of can be proved by showing density of , the collection of periodic orbits. In the proof that periodic orbits are dense in , most steps were standard manipulations that work for any topological dynamical system: decomposing a measure near into a convex combination of ergodic measures , finding generic points for each , and approximating with a -measure supported on the orbits of the . The only place where we needed the system to be the full shift was when we took a collection of words, representing orbit segments, and concatenated them to obtain a periodic orbit that shadows each of those orbit segments in turn.
Two problems arise when we try to generalise this proof to arbitrary transitive systems. Transitivity guarantees the existence of such a shadowing orbit, but
- the shadowing orbit need not be periodic, and
- we may need to insert arbitrarily long gaps between the segments being shadowed.
The first of these means that it is not clear how to obtain an ergodic measure, since an orbit segment that is not a complete periodic orbit does not support an invariant measure. The second of these means that even if we obtain an ergodic measure for which the shadowing orbit is generic, we will not be able to relate the integrals to the averages of along the orbit segments , which was necessary to show that is close to the original measure . The problem is that because we have no control over the gap length , the orbit segments along which we know what is happening may turn out to be statistically insignificant; this happens if .
Both of these problems vanish if we replace transitivity with the specification property, introduced by Bowen in 1971, and this is the idea behind Sigmund’s approach.
Definition 3 has specification if the conclusion of Proposition 2 holds uniformly — that is, for every there exists such that for every , the intersection in (1) is non-empty with for all , and moreover, the intersection contains a periodic point of period .
Now we can state the relevant result from Sigmund’s paper.
Theorem 4 If is a compact metric space and is a continuous map with specification, then is dense in .
The idea is simple enough: although we cannot immediately jump from one orbit segment to the next, we can do so with a uniformly bounded gap, and since remains fixed as the get larger and larger, its effect on the statistical properties of the shadowing orbit is negligible. Since the shadowing orbit is periodic, we are able to find a periodic orbit measure near any invariant measure.
To summarise: If has the specification property, then is the Poulson simplex. In particular, is dense and arc-connected.
Two questions immediately present themselves.
- Which dynamical systems satisfy the specification property, so that we can apply the above result?
- How tight is this result? Are there weaker properties that would do the job?
In future posts, we’ll see that while important classes of systems have specification (transitive Axiom A maps, for example), there are many interesting systems for which specification fails, but some weaker form of specification holds and allows us to obtain similar results.