In this post I discuss the Ledrappier-Walters formula.

Set Up

EDIT: This post is still unfinished

Throughout, let \((X,d)\) be a compact metric space and \(T : X \rightarrow X\) a continuous map (which is our dynamical system). For \(\delta > 0\) and \(n \in \mathbb{N}\) we say a set \(E \subseteq X\) is delta separated if for any \(x \neq y \in E\) we have

\[\max_{0 \leq i < n} d(T^i(x), T^i(y)) \geq \delta.\]

Let \(f : X \rightarrow \mathbb{R}\) be any function and define

\[f_n(x) := \sum_{i=0}^{n-1} f(T^i(x)).\]

We can define the exponential accumulation by

\[S(f, T, \delta, n) := \inf\left\{\sum_{x \in E} \exp(f_n(x)) : E \subseteq X \text{ is a delta separated set} \right\}.\]

The topological pressure is then defined as

\[P(f, T) := \lim_{\epsilon \rightarrow 0} \limsup_{n \rightarrow \infty} \frac{1}{n} \log(S(f,T,\delta,n)).\]

Note that we recover the entropy if we let \(f = 0\) in the above definition, so

\[h(T) = P(0,T).\]

One observation to make is that the space of \(T\)-invariant probability measures on \(X\) is compact (in the weak*-topology) and convex, with extreme points the ergodic measures (this can be found in Walters book).

The next goal is to define measure theoretic entropy. Recall that a measurable partition for a measure space \((X, \mathcal{M}, \mu)\) is a collection of measurable subsets \(\xi := \{A_i\}\) such that the following hold:

(1) \(\mu \left( X \setminus \left(\bigcup_i A_i \right) \right) = 0.\)

(2) \(\mu(A_i \cap A_j) = 0\) for all $i \neq j$$.

A measurable partition is then a collection of subsets such that they cover \(X\) almost everywhere and they are almost everywhere disjoint. We will make the additional assumption that \(\mu(A_i) > 0\) for each \(i\), which just generally means we ignore the sets of measure \(0\) in our calculations. All measurable partitions are going to be either finite or countable with finite entropy.

Let \(\xi\) be a measurable partition. We define the entropy of \(\xi\) to be

\[H_\mu(\xi) := \sum_{i \in I} \mu(A_i) \log(\mu(A_i)).\]

If one doesn’t want to ignore zero measure sets in their partition, then we use the same definition with the convention that \(0 \log(0) = 0\).

We define the conditional measure by

\[\mu(A \mid B) := \frac{\mu(A \cap B)}{\mu(B)}.\]

If we have two measurable partitions \(\xi := \{A_i\}_{i \in I}\) and \(\eta := \{B_j\}_{j \in J}\), then we define the conditional entropy of \(\xi\) with respect to \(\eta\) by

\[H_\mu(\xi \mid \eta) := -\sum_{j \in J} \mu(B_j) \sum_{i \in I} \mu(A_i \mid B_j) \log(\mu(A_i \mid B_j)).\]

We also define the join of two measurable partitions by

\[\xi \vee \eta := \{A \cap B : A \in \xi, B \in \eta\}.\]

If \(\xi\) is a measurable partition and \(T : X \rightarrow X\) a measure preserving transformation, then we define the joint partition by

\[\xi_{n}^T := \bigvee_{i=0}^{n-1} T^{-i}(\xi).\]

The measure entropy of a measure preserving transformation \(T : X \rightarrow X\) and a measurable partition \(\xi\) is defined by

\[h_\mu(T, \xi) := \lim_{n \rightarrow \infty} \frac{1}{n} H(\xi_{n}^T).\]

The measure entropy of just \(T\) is then defined as

\[h_\mu(T) := \sup\{h_\mu(T,\xi) : \xi \text{ is a finite measurable partition of }X\}.\]

Suppose now \(\pi : X \rightarrow Y\) is a measure preserving transformation of \(X\) onto a probability space \(Y\) so that \(\pi T = S \pi\) for a measure preserving transformation \(S : Y \rightarrow Y\). If \(\xi\) a measurable partition of \(X\), then we define the relative entropy of \(T\) with respect to \(S\) and the partition \(\xi\) as

\[h_\mu(T \mid S, \xi) := \lim_{n \rightarrow \infty} H_\mu(\xi_n^T \mid \pi^{-1}(\epsilon_Y)),\]

where \(\epsilon_Y\) is the partition of \(Y\) into points. We define the relative entropy of \(T\) with respect to \(S\) by

\[h_\mu(T \mid S) := \sup\{ h_\mu(T \mid S, \xi) : \xi \text{ is a finite measurable partition of } X\}.\]

Claim: Let \(\xi, \eta\) be finite partitions of \(X\). We have

\[h_\mu(T \mid S, \xi) \leq h_\mu(T \mid S, \eta) + H_\mu(\xi \mid \eta \vee \pi^{-1}(\epsilon_Y)).\]

Proof: Recall that we have the following nice properties for entropy.

(1) For any finite measurable partitions \(\xi, \eta\) we have

\[H_\mu(\xi \vee \eta) = H(\xi) + H(\eta \mid \xi).\]

(2) For any finite measurable partitions \(\xi, \eta, \rho\) we have

\[H_\mu(\eta \vee \xi \mid \rho) + H_\mu(\xi \mid \rho) + H(\eta \mid \xi \vee \rho).\]

Using the above, we see that for fixed \(n\) we have

\[H_\mu(\xi_n^T \mid \eta_n^T \vee \pi^{-1}(\epsilon_Y)) + H_\mu(\eta_n^T \mid \pi^{-1}(\epsilon_Y)) = H_\mu(\xi_n^T \vee \eta_n^T \mid \pi^{-1}(\epsilon_Y)) \geq H(\xi_n^T \mid \pi^{-1}(\epsilon_Y)).\]

To show our goal, we just need to study the term \(H_\mu(\xi_n^T \mid \eta_n^T \vee \pi^{-1}(\epsilon_Y))\). The trick is to proceed inductively. Notice

\[H_\mu(\xi_n^T \mid \eta_n^T \vee \pi^{-1}(\epsilon_Y)) = H_\mu(\xi \mid \eta_n^T \vee \pi^{-1}(\epsilon_Y)) + H_\mu(T^{-1}(\xi_{n-1}^T) \mid \xi \vee \eta_n^T \vee \pi^{-1}(\epsilon_Y)).\]

We can get an upper bound:

\[H_\mu(\xi_n^T \mid \eta_n^T \vee \pi^{-1}(\epsilon_Y)) \leq H_\mu(\xi \mid \eta \vee \pi^{-1}(\epsilon_Y)) + H_\mu(T^{-1}(\xi_{n-1}^T) \mid \eta_n^T \vee \pi^{-1}(\epsilon_Y)).\]

We can continue in this fashion to eventually get

\[H_\mu(\xi_n^T \mid \eta_n^T \vee \pi^{-1}(\epsilon_Y)) \leq n H_\mu(\xi \mid \eta \vee \pi^{-1}(\epsilon_Y)).\]

The result now follows by plugging in limits. See Katok and Hasselblatt Proposition 4.3.10 for more details. \(\blacksquare\)

We define the boundary of a measurable partition \(\xi := \{A_1, \ldots, A_n\}\) to be

\[\partial \xi := \bigcup_{i=1}^n \partial A_i.\]

We define the measure \(\delta_x\) to be the probability measure concentrated at the point \(x\) (that is, the Dirac measure). Also recall that if \(\pi : X \rightarrow Y\) a measurable transformation, \(\mu\) a measure on \(X\), then the pushforward measure is defined by

\[\pi^*(\mu)(E) = \mu(\pi^{-1}(E)).\]

Finally if \(\pi : X \rightarrow Y\) is continuous, \(T : X \rightarrow X\) is continuoius, \(S : Y \rightarrow Y\) is continuous, and \(\pi T = S \pi\), then we say that \(S\) is a factor of \(T\).

Theorem

With all of this, we can now actually state the major theorem (see Theorem 2.1 in the paper).

Theorem: Let \(X, Y\) be compact metric spaces and let \(T : X \rightarrow X\), \(S : Y \rightarrow Y\), \(\pi : X \rightarrow Y\) be continuous maps so that \(\pi\) is surjective and \(\pi T = S \pi\). Let \(f : X \rightarrow \mathbb{R}\) be continuous and let \(\nu \in M(Y,S)\). Then

\[\sup \left\{h_\mu(T \mid S) + \int f d\mu : \mu \in M(X,T) \text{ and } \pi^*(\mu) = \nu\right\} = \int_Y P(T,f,\pi^{-1}(y))d\nu(y).\]

Sketch of Proof

Throughout the proof, we assume the conditions of the theorem above, namely we assume that \(X, Y\) are compact metric spaces, \(T :X \rightarrow X\), \(S : Y \rightarrow Y\), \(\pi : X \rightarrow Y\) are all continuous, \(\pi\) is surjective, and \(S\) is a factor of \(T\). The first lemma says that conditional entropy with respects to maps behaves similar to conditional entropy with respect to partitions. This is Lemma 3.1 in the paper.

Lemma: Let \(Z\) be a compact space and \(R\) a factor of \(S\) by \(\delta : Y \rightarrow Z\). If \(\mu \in M(X,T)\) then we have

\[h_\mu(T \mid R) = h_{\pi^*(\mu)}(S \mid R) + h_\mu(T \mid S).\]

The proof of this essentially follows from the so-called Pinsker’s lemma (see Rokhlin’s lecture notes). While a little technical, it is essentially the same as it’s partition counterpart. The next lemma is also important, but a little bit technical. This is Lemma 3.2 in the paper.

Lemma: (i) Let \(\xi, \eta\) be finite partitions of \(X\), let \(\mu_1, \mu_2\) be two probability measures on Borel subsets of \(X\). If \(0 \leq p \leq 1\) then

\[p H_{\mu_1}(\xi \mid \eta\) + (1-p) H_{\mu_2} (\xi \mid \eta) \leq H_{p \mu_1 + (1-p)\mu_2}(\xi \mid \eta).\]

(ii) Suppose \(\xi\) is a finite partition of \(X\) and let \(\mu\) be a finite Borel probability measure on \(X\) with \(\mu(\partial \xi) = 0\). Let \(\{\mu_n\}\) be a sequence of Borel probability measures so that \(\mu_n \rightarrow \mu\). Then

\[\lim_{n \rightarrow \infty} H_{\mu_n}(\xi \mid \eta) \leq H_\mu(\xi \mid \eta).\]

(iii) The function \(\mu \mapsto h_\mu(T \mid S)\) defined on \(M(X,T)\) is affine.

We omit the proof of this one as well. Observe that the point of the prior lemma was to be used for this lemma. With this in mind, if we let \(\|f\|\) denote the supremum norm for \(C(X)\) then we can set up the following lemma (this is Lemma 3.3 in the paper).

Lemma: Let \(f \in C(X)\). The mappings \(y \mapsto P_n(T, f, \pi^{-1}(y), \delta)\) of \(Y\) to \(\mathbb{R}\) are Borel measurable for \(n \geq 1\). Moreover for each \(\delta > 0\) there is a constant \(C(\delta)\) so that

\[\frac{1}{n} \log(P_n(T, f, \pi^{-1}(y), \delta)) < C(\delta) \text{ for } n \geq 1, y \in Y.\]

Sketch of Proof: Recall that

\[P_n(T,f,\pi^{-1}(y), \delta) := \sup \left\{ \sum_{x \in E} \exp(f_n(x)) : E \text{ is } (n,\delta) \text{ separated and } E \subseteq \pi^{-1}(y) \right\}.\]

Ledrappier-Walters breaks this up into smaller more manageable chunks which are easy to study. The idea is to fix \(k \in \mathbb{N}\) and \(\delta > 0\) and look at those chunks which have \(k\) factors that stay roughly larger than \(e^{nt}\) for some \(t \in \mathbb{R}\). The first step is to look at the collection of \(k\)-tuples of points in \(X\) who all project down into \(y\). That is, we look at

\[D_k := \{(x_1, \ldots, x_k) \in X^k : \pi(x_1) = \cdots = \pi(x_k)\}.\]

Now we want them to stay separated, so look at

\[E_k^{n,\delta} := \{(x_1, \ldots, x_k) \in X^k : \max_{0 \leq m < n} d(T^m(x_i), T^m(x_j)) \geq \delta \text{ if } i \neq j\}.\]

Finally we want this approximately larger than the exponential sum, so

\[F_{k,l}^{n,t} := \left\{ (x_1, \ldots, x_k) \in X^k : \sum_{i=1}^k \exp(f_n(x_i)) \geq e^{nt} + 1/l\right\}.\]

The advantage now is that each of these sets are going to be closed. The first one is the preimage of a closed set under a continuous map, the second is the intersection of a bunch of closed sets, and the third one is the preimage of a closed set under a continuous map. Let \(\phi : X^k \rightarrow Y, \phi(x_1, \ldots, x_k) = \pi(x_1)\). Taking the intersection and mapping it through \(\phi\), we have

\[G_{k,l}^{n,\delta,t} = \phi(D_k \cap E_k^{n,\delta} \cap F_{k,l}^{n,t}).\]

This is a compact set, and furthermore these are the slices (in terms of \(k\) and \(l\)) of things we actually care about. Taking the union over all \(k\) and \(l\) gives us a Borel set which is exactly

\[\bigcup_{k,l} G_{k,l}^{n,\delta,t} = \{y : P_n(T, f, \pi^{-1}(y), \delta) > e^{nt}\}.\]

This tells us the mapping is Borel. If one makes the observation that \(P_n(T,f,\pi^{-1}(y), \delta) \leq e^{n \|f\|} s_n(T,\pi^{-1}(y), \delta),\) then the last condition follows easily. \(\blacksquare\)

The final lemma is one that should seem familiar from ergodic theory. This is Lemma 3.4 in the paper.

Lemma: If \(K \subseteq X\) compact and \(m \in \mathbb{N}\) then

\[P(T^m, f_m, K) = m P(T,f,K).\]

Sketch of Proof: It’s mostly just a matter of running through definitions. Suppose \(E\) is \((n,\delta)\) separated for \(T^m\). This means that for \(x \neq y\) in \(E\) we have

\[\max_{0 \leq i < nm} d(T^i(x), T^i(y)) \geq \max_{0 \leq k < n} d(T^{km}(x), T^{km}(y)) \geq \delta.\]

Thus \(E\) is an \((mn,\delta)\) separated set for \(T\). Now

\[P_n(T^m, f_m, K, \delta) \leq P_{nm}(T,f,K, \delta).\]

Thus

\[\frac{1}{n} \log(P_n(T^m, f_m, K, \delta)) \leq m \cdot \frac{1}{nm} \log(P_{nm}(T,f,K,\delta)).\]

Take the limit as \(n \rightarrow \infty\) and \(\delta \rightarrow 0\) to get the inequality going in one direction. The other direction follows from a clever choice of \(\delta\) and \(\delta'\) and then doing the same kind of inequality tricks. \(\blacksquare\)