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

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

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

**1. Definitions and result **

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

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

and in general

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

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

where . We write

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

Theorem 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 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 2Let be a topologically mixing Anosov diffeomorphism on and the quantities above. LetGiven 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 3Let be a compact manifold and a topologically mixing Anosov diffeomorphism. For every there are constants and such that for every -Hölder potential , we haveso 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

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 , there is such 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 . 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.