Entropy, volume growth and SRB measures for Banach space mappings

Abstract

We consider $C^2$ Fréchet differentiable mappings of Banach spaces leaving invariant compactly supported Borel probability measures, and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces. SRB measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds, equivalently the sum of positive Lyapunov exponents of the map. In addition to numerous difficulties incurred by our infinite-dimensional setting, a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions. The results here generalize previously known results for diffeomorphisms of finite dimensional Riemannian manifolds, and are applicable to dynamical systems defined by large classes of dissipative parabolic PDEs.

Appendix

Recall that the hypothesis of Lemma 3.3 are

$$\begin{aligned} (*) \quad E, E', F \in \mathcal {G}(\mathcal {B}), \quad \mathcal {B}= E \oplus F, \quad \text{ and } \quad d_H(E, E') < |\pi _{E //F}|^{-1}. \end{aligned}$$

Let us write $d(\cdot , \cdot )$ instead of $d_H(\cdot , \cdot )$ for simplicity.

Proof of Lemma 3.3

To prove $\mathcal {B}= E' \oplus F$, we first show $E' \cap F = \{0\}$. If not, pick $e' \in E' \cap F$ with $|e'|=1$. Since $d(e', S_E) \le d(E, E') < |\pi _{E //F}|^{-1}$, there exists $e \in E$ with $|e|=1$ such that $|e-e'| < |\pi _{E //F}|^{-1}$. This is incompatible with

$$\begin{aligned} 1 = |e| = |\pi _{E //F} e| = |\pi _{E //F} (e - e')| \le |\pi _{E //F}| \cdot |e - e'| < 1. \end{aligned}$$

Next, we claim that $E' \oplus F$ is closed. It will suffice to show that there exists $A > 0$ such that for any $e' \in E', f \in F$, we have

$$\begin{aligned} |e'| \le A |e' + f|. \end{aligned}$$

(37)

This is known as the Kober criterion [15]. Indeed, if (37) holds and $x_n = e'_n + f_n$ is Cauchy, then $e_n'$ and $f_n$ individually are Cauchy, and thus converge to some $e' \in E', f \in F$ respectively, hence $x_n \rightarrow x := e' + f \in E' + F$. To prove (37), pick arbitrary $e' \in E'$ and $f \in F$, and fix $c>1$ with $c d(E,E')< |\pi _{E //F}|^{-1}$. As before, let $e \in E$ be such that $|e| = |e'|$ and $|e - e'| \le |e'| c d(E, E')$. Then

$$\begin{aligned} |e' + f| \ge |e + f| - |e - e'| \ge |e'| (|\pi _{E //F}|^{-1} - c d(E, E')) =: A^{-1} |e'|. \end{aligned}$$

To finish, assume for the sake of contradiction that $E' \oplus F \ne \mathcal {B}$. By Assumption (ii), there exist $c_1<1<c_2$ such that $c_2 d(E,E') |\pi _{E //F}| < c_1$. Since $E' \oplus F$ is closed, the Riesz Lemma [39] asserts that there exists $x \in \mathcal {B}$ with $|x|=1$ such that $|x - (e' + f)| \ge c_1$ for all $e' \in E', f \in F$. On the other hand, since $\mathcal {B}= E \oplus F$, we have that $x = e + f$ for some $e \in E, f \in F$; notice that $|e| \le |\pi _{E //F}|$. But there exists $e' \in E$ with $|e'|=|e|$ and $|e-e'| \le c_2 |e| d(E,E')$, and for such an $e'$,

$$\begin{aligned} |x - (e' + f)| = |e - e'| \le |e| c_2 d(E, E') < c_1, \end{aligned}$$

contradicting our choice of x.

Lemma A.1. Assume (*). Then

• (i)
  $$\begin{aligned} |\pi _{E' //F}| \le \frac{|\pi _{E //F}|}{1 - |\pi _{E //F}| d(E, E')}, \end{aligned}$$
• (ii)
  $$\begin{aligned} |\pi _{F //E'}|_E| \le 2 |\pi _{E' //F}| d(E, E') \, . \end{aligned}$$

Proof

Since $\mathcal {B}= E\oplus F = E'\oplus F$, (i) above is equivalent to

$$\begin{aligned} \alpha (E, F) \le d(E, E') + \alpha (E', F) \end{aligned}$$

(38)

by the formula $\alpha (E, F) = |\pi _{E //F}|^{-1}$ from Sect. 3.1.2. To estimate $\alpha (E', F)$ from below, we let $e' \in E'$ with $|e'| = 1$ and $f \in F$ be arbitrary. For $c > 1$, we let $e \in E, |e| = 1$ be such that $|e - e'| \le c d(E, E')$. Then,

$$\begin{aligned} |e' - f| \ge |e - f| - |e' - e| \ge \alpha (E, F) - c d(E, E'). \end{aligned}$$

But $e', f$ were arbitrary and so our formula follows on taking $c \rightarrow 1$.

To prove (ii), fix $e \in E, |e| = 1$. Then for $c > 1$ arbitrarily close to 1, let $e' \in E', |e'| = 1$ be such that $|e - e'| \le c d(E, E')$. Then

$$\begin{aligned} |\pi _{F //E'} e| = |\pi _{F //E'} (e - e')| \le |\pi _{F //E'}| \cdot |e - e'| \le 2 |\pi _{E' //F}| \cdot c d(E, E'). \end{aligned}$$

$\square$

Proof of Lemma 3.5

That $\mathcal {B}= E'\oplus F$ follows from Lemma 3.3. The bounds in (a) are given by Lemma A.1, and (b) follows from (a).