Let be a compact metric space and a homeomorphism. Recall that an equilibrium state for a continuous potential function is an -invariant Borel probability measure on maximizing the quantity over all invariant probabilities; the topological pressure is the value of this maximum.
A classical result on existence and uniqueness of equilibrium states is due to Bowen, who proved that if is expansive and has specification, and has a bounded distortion property (the `Bowen property’), then there is a unique equilibrium state . In particular, this applies when is Anosov and is Hölder.
It seems to be well-known among experts that under Bowen’s hypotheses, must have positive entropy (equivalently, ), but I do not know of an explicit reference. In this post I provide a proof of this fact, which also gives reasonably concrete bounds on the entropy of ; equivalently, a bound on the size of the gap .
1. Definitions and result
First, let’s recall the definitions in the form that I will need them. Given , , and , the Bowen ball around of order and radius is the set
The map has specification if for every there is such that for every and , there is such that
and in general
for every . We refer to as the “gluing time”; one could also consider a weaker property where the gluing times are allowed to vary but must be bounded above by ; this makes the estimates below more complicated, so for simplicity we will stick with the stronger version.
A function has the Bowen property at scale with distortion constant if is such that
where . We write
where is the collection of -separated subsets of (those sets for which whenever , ). The topological pressure is , where
In particular, if is an equilibrium state for , then we have .
2. Consequence for Anosov diffeomorphisms
Before proving the theorem we point out a useful corollary. If is a compact manifold and is a topologically mixing Anosov diffeomorphism, then has specification at every scale (similar results apply in the Axiom A case). Moreover, every Hölder continuous potential has the Bowen property, and thus Theorem 1 applies.
For an Anosov diffeo, the constants and in (1) can be controlled by the following factors (here we fix a small ):
- 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 Hölder exponent () and constant () for the potential .
For the specification property for an Anosov diffeo, is determined by the condition that , so that small pieces of unstable manifold expand to become -dense within iterates; thus we have
For the Bowen property, one compares and by comparing each to , where is the (Smale bracket) intersection point coming from the local product structure. Standard estimates give , so the Hölder property gives
A similar estimate for gives
Thus Theorem 1 has the following consequence for Anosov diffeomorphisms.
Corollary 2 Let be a topologically mixing Anosov diffeomorphism on and the quantities above. Let
Given a -Hölder potential , consider the quantity
Then we have
so that in particular, if is an equilibrium state for , then
Finally, note that since shifting the value of by a constant does not change its equilibrium states, we can assume without loss of generality that and write the following consequence of the above, which is somewhat simpler in appearance.
Corollary 3 Let be a compact manifold and a topologically mixing Anosov diffeomorphism. For every there are constants and such that for every -Hölder potential , we have
so that as before, if is an equilibrium state for , we have
This corollary gives a precise bound on how the entropy of a family of equilibrium states can decay as the Hölder semi-norms of the corresponding potentials become large. To put it another way, given any threshold , this gives an estimate on how large must be before can have an equilibrium state with entropy below .
3. Proof of the theorem
We spend the rest of the post proving Theorem 1. Fix and consider for each the orbit segment . Fix such that . 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 . Integrating over and gives
and thus we have
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
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 , there is such that and ; since this guarantees 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 . Note that we must have in order for our construction to work.
A straightforward differentiation and some routine algebra shows that occurs when , at which point we have .
For this value of to lie in , we must have . If , then achieves its maximum when , and again some routine algebra shows that at this point we have . Taken together, these two estimates prove Theorem 1.