Entropy bounds for equilibrium states

[Update 6/15/17: The original version of this post had a small error in it, which has been corrected in the present version; the definition of {\mathcal{I}_n} in the proof of the main theorem needed to be modified so that each {k_i} is a multiple of {2(\tau+1)}.  Thanks to Leonard Carapezza for pointing this out to me.]

Let {X} be a compact metric space and {f\colon X\rightarrow X} a homeomorphism. Recall that an equilibrium state for a continuous potential function {\varphi\colon X\rightarrow {\mathbb R}} is an {f}-invariant Borel probability measure on {X} maximizing the quantity {h_\mu(f) + \int\varphi\,d\mu} over all invariant probabilities; the topological pressure {P(\varphi)} is the value of this maximum.

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

It seems to be well-known among experts that under Bowen’s hypotheses, {\mu_\varphi} must have positive entropy (equivalently, {P(\varphi) > \sup_\mu \int\varphi\,d\mu}), 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 {\mu_\varphi}; equivalently, a bound on the size of the gap {P(\varphi) - \sup_\mu \int\varphi\,d\mu}.

1. Definitions and result

First, let’s recall the definitions in the form that I will need them. Given {x\in X}, {n\in {\mathbb N}}, and {\varepsilon>0}, the Bowen ball around {x} of order {n} and radius {\varepsilon} is the set

\displaystyle B_n(x,\varepsilon) := \{y\in X : d(f^kx, f^ky) < \varepsilon \text{ for all } 0\leq k < n\}.

The map {f} has specification if for every {\varepsilon>0} there is {\tau=\tau(\varepsilon)\in {\mathbb N}} such that for every {x_1,\dots, x_k\in X} and {n_1,\dots, n_k\in {\mathbb N}}, there is {x\in X} such that

\displaystyle x\in B_{n_1}(x_1,\varepsilon),\qquad f^{n_1 + \tau}(x)\in B_{n_2}(x_2,\varepsilon),

and in general

\displaystyle f^{n_1 + \tau + \cdots + n_{i-1} + \tau}(x) \in B_{n_i}(x_i,\varepsilon)

for every {1\leq k\leq n}. We refer to {\tau} 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 {\tau}; this makes the estimates below more complicated, so for simplicity we will stick with the stronger version.

A function {\varphi\colon X\rightarrow {\mathbb R}} has the Bowen property at scale {\varepsilon} with distortion constant {V} if {V\in {\mathbb R}} is such that

\displaystyle |S_n\varphi(x) - S_n\varphi(y)| \leq V \text{ for all } x\in X\text{ and } y\in B_n(x,\varepsilon),

where {S_n\varphi(x) := \sum_{k=0}^{n-1} \varphi(f^k x)}. We write

\displaystyle \Lambda_n(\varphi,\varepsilon) := \sup_{E\in \mathcal{E}_{n,\varepsilon}} \sum_{x\in E} e^{S_n\varphi(x)},

where {\mathcal{E}_{n,\varepsilon}} is the collection of {(n,\varepsilon)}-separated subsets of {X} (those sets {E\subset X} for which {y\notin B_n(x,\varepsilon)} whenever {x,y\in E}, {x\neq y}). The topological pressure is {P(\varphi) = \lim_{\varepsilon\rightarrow 0} P(\varphi,\varepsilon)}, where

\displaystyle P(\varphi,\varepsilon) = \limsup_{n\rightarrow\infty} \frac 1n \log \Lambda_n(\varphi,\varepsilon).

Theorem 1 Let {X} be a compact metric space with diameter {>6\varepsilon}, {f\colon X\rightarrow X} a homeomorphism with specification at scale {\varepsilon} with gap size {\tau}, and {\varphi\colon X\rightarrow {\mathbb R}} a potential with the Bowen property at scale {\varepsilon} with distortion constant {V}. Let

\displaystyle \Delta = \frac{\log(1+e^{-(V+2(2\tau+1)\|\varphi\|)})}{2(\tau+1)}

where {\|\varphi\| = \sup_{x\in X} |\varphi(x)|}. Then we have

\displaystyle P(\varphi) \geq P(\varphi,\varepsilon) \geq \Big( \sup_\mu \int\varphi\,d\mu\Big) + \Delta. \ \ \ \ \ (1)

 

In particular, if {\mu} is an equilibrium state for {\varphi}, then we have {h_\mu(f) \geq \Delta > 0}.

2. Consequence for Anosov diffeomorphisms

Before proving the theorem we point out a useful corollary. If {M} is a compact manifold and {f\colon M\rightarrow M} is a topologically mixing {C^1} Anosov diffeomorphism, then {f} 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 {V} and {\tau} in (1) can be controlled by the following factors (here we fix a small {\varepsilon>0}):

  1. the rate of expansion and contraction along the stable and unstable directions, given in terms of {C,\lambda>0} such that {\|Df^n_x(v^s)\| \leq C e^{-\lambda n}} for all {n\geq 0} and {v^s\in E^s}, and similarly for {v^u\in E^u} and {n\leq 0};
  2. how quickly unstable manifolds become dense, in other words, the value of {R>0} such that {W_R^u(x)} is {\varepsilon}-dense for every choice of {x};
  3. the angle between stable and unstable directions, which controls the local product structure, in particular via a constant {K>0} such that {d(x,y) < \varepsilon} implies that {W_{K\varepsilon}^s(x)} intersects {W_{K\varepsilon}^u(y)} in a unique point {z}, and the leafwise distances from {x,y} to {z} are at most {K d(x,y)};
  4. the Hölder exponent ({\beta}) and constant ({|\varphi|_\beta}) for the potential {\varphi}.

For the specification property for an Anosov diffeo, {\tau =\tau(\varepsilon)} is determined by the condition that {C^{-1}e^{\lambda\tau}(\varepsilon/K) > R}, so that small pieces of unstable manifold expand to become {\varepsilon}-dense within {\tau} iterates; thus we have

\displaystyle \tau(\varepsilon) \approx \lambda^{-1} \log(R(\varepsilon) KC\varepsilon^{-1}).

For the Bowen property, one compares {S_n\varphi(x)} and {S_n\varphi(y)} by comparing each to {S_n\varphi(z)}, where {z} is the (Smale bracket) intersection point coming from the local product structure. Standard estimates give {d(f^j x, f^jz) \leq CK\varepsilon e^{-\lambda j}}, so the Hölder property gives

\displaystyle \begin{aligned} |S_n\varphi(x) - S_n\varphi(z)| &\leq \sum_{j=0}^{n-1} |\varphi(f^j x) - \varphi(f^j z)| \leq \sum_{j=0}^{n-1} |\varphi|_\beta d(f^jx,f^jz)^\beta \\ &\leq |\varphi|_\beta \sum_{j=0}^\infty (CK\varepsilon)^\beta e^{-\lambda\beta j} = |\varphi|_\beta (CK\varepsilon)^\beta (1-e^{-\lambda\beta})^{-1}. \end{aligned}

A similar estimate for {|S_n\varphi(y) - S_n\varphi(z)|} gives

\displaystyle V = 2(CK\varepsilon)^\beta(1- e^{-\lambda\beta})^{-1} |\varphi|_\beta.

Thus Theorem 1 has the following consequence for Anosov diffeomorphisms.

Corollary 2 Let {f} be a topologically mixing Anosov diffeomorphism on {M} and {C,\lambda,\varepsilon,R,K} the quantities above. Let

\displaystyle \delta = \frac{\lambda}{2\log(RKC\varepsilon^{-1})}.

Given a {\beta}-Hölder potential {\varphi\colon M\rightarrow {\mathbb R}}, consider the quantity

\displaystyle Q(\varphi) := 2(CK\varepsilon)^\beta (1-e^{-\lambda\beta})^{-1} |\varphi|_\beta + 5\lambda^{-1} \log(RKC\varepsilon^{-1})\|\varphi\|.

Then we have

\displaystyle P(\varphi) \geq P(\varphi,\varepsilon) \geq \Big(\sup_\mu \int\varphi\,d\mu\Big) + \delta \log(1+e^{-Q(\varphi)})

so that in particular, if {\mu} is an equilibrium state for {\varphi}, then

\displaystyle h_\mu(f) \geq \delta \log(1+e^{-Q(\varphi)}) > 0.

Finally, note that since shifting the value of {\varphi} by a constant does not change its equilibrium states, we can assume without loss of generality that {\|\varphi\| \leq (\mathrm{diam}\, M)^\beta |\varphi|_\beta} and write the following consequence of the above, which is somewhat simpler in appearance.

Corollary 3 Let {M} be a compact manifold and {f\colon M\rightarrow M} a topologically mixing Anosov diffeomorphism. For every {\beta>0} there are constants {\delta = \delta(M,f)>0} and {R = R(M,f,\beta)} such that for every {\beta}-Hölder potential {\varphi}, we have

\displaystyle P(\varphi) \geq \Big(\sup_\mu \int\varphi\,d\mu\Big) + \delta e^{-R|\varphi|_\beta}

so that as before, if {\mu} is an equilibrium state for {\varphi}, we have

\displaystyle h_\mu(f) \geq \delta e^{-R|\varphi|_\beta} > 0.

This corollary gives a precise bound on how the entropy of a family of equilibrium states can decay as the Hölder semi-norms {|\varphi|_\beta} of the corresponding potentials become large. To put it another way, given any threshold {h_0>0}, this gives an estimate on how large {|\varphi|_\beta} must be before {\varphi} can have an equilibrium state with entropy below {h_0}.

3. Proof of the theorem

We spend the rest of the post proving Theorem 1. Fix {x\in X} and consider for each {n\in {\mathbb N}} the orbit segment {x, f(x), \dots, f^{n-1}(x)}. Fix {\alpha\in (0,\frac 12]}. Let {m_n = \lceil \frac{\alpha n}{2(\tau+1)} \rceil}, and let

\displaystyle \mathcal{I}_n = \{ 0 < k_1 < k_2 < \cdots < k_{m_n} < n : k_i \in 2(\tau+1){\mathbb N} \ \forall i\}.

Write {k_0 = 0} and {k_{m_n + 1} = n}. The idea is that for each {\vec k\in \mathcal{I}_n}, we will use the specification property to construct a point {\pi(\vec k) \in X} whose orbit shadows the orbit of {x} from time {0} to time {n}, except for the times {k_i}, at which it deviates briefly; thus the points {\pi(\vec k)} will be {(n,\varepsilon)}-separated on the one hand, and on the other hand will have ergodic averages close to that of {x}.

First we estimate {\#\mathcal{I}_n} from below; this requires a lower bound on {{n\choose \ell}}. Integrating {\log t} over {[1,k]} and {[1,k+1]} gives

\displaystyle k\log k - k + 1 \leq \log(k!) \leq k\log k - k + 1 + \log(k+1),

and thus we have

\displaystyle \begin{aligned} \log{n\choose \ell} &= \log(n!) - \log(\ell!) - \log(n-\ell)! \\ &\geq n\log n + 1 - \ell\log\ell - (n-\ell)\log(n-\ell) - \log((\ell+1)(n-\ell+1)) \\ &\geq h\big( \tfrac\ell n\big) n - 2\log n, \end{aligned}

where {h(\delta) = -\delta\log\delta - (1-\delta)\log(1-\delta)}. This function is increasing on {(0,\frac12)}, so

\displaystyle \begin{aligned} \log\#\mathcal{I}_n &\geq \log{\lfloor \frac{n}{2(\tau+1)}\rfloor \choose m_n} \geq h(\tfrac {2(\tau+1) m_n} n) \frac{n}{2(\tau+1)} - 2\log \frac{n}{2(\tau+1)} \\ &\geq \frac{h(\alpha)}{2(\tau+1)} n - 2\log n. \end{aligned} \ \ \ \ \ (2)

 

Given {k\in \{0, \dots, n-1\}}, let {y_k \in X} be any point with {d(f^k(x),y_k) > 3\varepsilon} (using the assumption on the diameter of {X}). Now for every {\vec{k}\in \mathcal{I}_n}, the specification property guarantees the existence of a point {\pi(\vec{k})\in X} with the property that

\displaystyle \begin{aligned} \pi(\vec{k}) &\in B_{k_1-\tau}(x,\varepsilon), \\ \qquad f^{k_1}(\pi(\vec{k})) &\in B(y_{k_1},\varepsilon), \\ \qquad f^{k_1+\tau+1}(\pi(\vec{k})) &\in B_{k_2 - k_1 - 2\tau - 1}(f^{k_1 + \tau+1}(x)), \end{aligned}

and so on, so that in general for any {0\leq i \leq m_n} we have

\displaystyle \begin{aligned} f^{k_i + \tau + 1}(\pi(\vec{k})) &\in B_{k_{i+1} - k_i - 2\tau - 1}(f^{k_i + \tau _ 1}(x)), \\ f^{k_{i+1}}(\pi(\vec{k})) &\in B(y_{k_{i+1}},\varepsilon). \end{aligned} \ \ \ \ \ (3)

 

Write {j_i = k_{i+1} - k_i - 2\tau - 1}; then the first inclusion in (3), together with the Bowen property, gives

\displaystyle |S_{j_i} \varphi(f^{k_i + \tau + 1}(x)) - S_{j_i} \varphi(f^{k_i + \tau + 1} (\pi (\vec{k}))| \leq V.

Now observe that for any {y\in X} we have

\displaystyle \bigg| S_n \varphi(y) - \sum_{i=0}^{m_n} S_{k_{i+1} - k_i - 2\tau-1}\varphi(f^{k_i + \tau + 1} y) \bigg| \leq (2\tau + 1)m_n \|\varphi\|.

We conclude that

\displaystyle |S_n\varphi(\pi(\vec{k})) - S_n\varphi(x)| \leq m_n(V + 2(2\tau+1)\|\varphi\|). \ \ \ \ \ (4)

 

Consider the set {\pi(\mathcal{I}_n) \subset X}. The second inclusion in (3) guarantees that this set is {(n,\varepsilon)}-separated; indeed, given any {\vec{k} \neq \vec{k}' \in \mathcal{I}_n}, we can take {i} to be minimal such that {k_i \neq k_i'}, let {j=k_i'}, and then observe that {f^j(\pi(\vec{k})) \in B(y_j, \varepsilon)} and {f^j(\pi(\vec{k}')) \in B(f^j(x),\varepsilon)}; since {d(y_j,f^j(x)) > 3\varepsilon} this guarantees that {\pi(\vec{k}') \notin B_n(\pi(\vec{k}),\varepsilon)}.

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

\displaystyle \begin{aligned} \Lambda_n(\phi,\varepsilon) &\geq \sum_{\vec{k} \in \pi(\mathcal{I}_n)} e^{S_n\varphi(\pi(\vec{k}))} \\ &\geq (\#\mathcal{I}_n) \exp\big(S_n\varphi(x) - m_n(V+2(2\tau+1)\|\varphi\|)\big) \\ &\geq n^{-2} \exp\big(S_n \varphi(x) + \tfrac {h(\alpha)}{2(\tau+1)} n - (\tfrac{\alpha}{2(\tau+1)} n + 1)(V+2(2\tau+1)\|\varphi\|)\big). \end{aligned}

Taking logs, dividing by {n}, and sending {n\rightarrow\infty} gives

\displaystyle P(\varphi,\varepsilon) \geq \Big(\limsup_{n\rightarrow\infty} \frac 1n S_n\varphi(x) \Big) + \frac 1{2(\tau+1)} \Big(h(\alpha) - \alpha(V+2(2\tau+1)\|\varphi\|) \Big).

Given any ergodic {\mu}, we can take a generic point {x} for {\mu} and conclude that the lim sup in the above expression is equal to {\int\varphi\,d\mu}. Thus to bound the difference {P(\varphi,\varepsilon) - \int\varphi\,d\mu}, we want to choose the value of {\alpha \in (0,\frac 12]} that maximizes {h(\alpha) - \alpha Q}, where {Q=V+2(2\tau+1)\|\varphi\|}.

A straightforward differentiation and some routine algebra shows that {\frac d{d\alpha} (h(\alpha) - \alpha Q) = 0} occurs when {\alpha = (1+e^Q)^{-1}}, at which point we have {h(\alpha) - \alpha Q = \log(1+e^{-Q})}, proving Theorem 1.

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About Vaughn Climenhaga

I'm an assistant professor of mathematics at the University of Houston. I'm interested in dynamical systems, ergodic theory, thermodynamic formalism, dimension theory, multifractal analysis, non-uniform hyperbolicity, and things along those lines.
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