Entropy of Physical Measures for $C^\infty$ Dynamical Systems

Abstract

For a $C^\infty$ map f on a compact manifold M we prove that for a Lebesgue randomly picked point x there is an empirical measure from x with entropy larger than or equal to the top Lyapunov exponent of $\Lambda \, df:\Lambda \,TM\circlearrowleft$ at x. This contrasts with the well-known Ruelle inequality. As a consequence we give some refinement of Tsujii’s work [23] relating physical and Sinai-Ruelle-Bowen measures.

Appendices

Appendix A. Counter-Example for $C^r$ Interval Maps for Any Finite r

For any positive integer r we give an example of a $C^r$ (but not $C^{r+1}$) interval map $h:[0,3/2]\circlearrowleft$ such that for x in a positive Lebesgue measure set the following properties hold:

• (1) the empirical measures $(\mu _n^x)_n$ are converging to the Dirac measure at a fixed point (therefore with zero entropy),

• (2) the Lyapunov exponent at x satisfies $\chi (x)=\frac{\log \Vert h'\Vert _\infty }{r}>0$.

Consequently the Main Theorem does not hold true in finite smoothness.

Step 1 Let $\lambda >1$. We first consider a $C^r$ (even $C^\infty$) interval map $f:[0,3/2]\circlearrowleft$ with the following properties

• $f(0)=f(1)=0$,

• f has a tangency of order r at 1, i.e. $f^{(k)}(1)=0$ for $k=1,\ldots ,r$,

• f is affine with a slope equal to $\lambda =\Vert f'\Vert _{\infty }$ on the interval $[0, 1/\lambda ]$.

Step 2 After a small $C^\infty$ perturbation of f around 1 we may build a new map g such that for some $n_0$ and $n\ge n_0$, $g^k(1-1/n)$ lies in $[0, 1/\lambda ]$ for $k=1,\ldots ,r^n-1$ and $g^{r^{n}}(1-1/n)=1-1/n+1$. Indeed these conditions require $g(1-1/n)=(1-1/n+1)\lambda ^{-r^n+1}=o(1/n^r)$, so that one can choose g arbitrarily $C^\infty$ closed to f by taking $n_0$ large enough. For the interval map g, the empirical measures at $1-1/n$ are converging to the Dirac measure at the fixed point 0. We may also assume g is constant on $J_n:=[1-1/n, 1-1/n-1/2n^2]$ for $n\ge n_0$.

Fig. 1

The graph ofg The arrows and points in blue represent the orbit of $1-1/n\in J_n$

Step 3 We lastly modify g on $J_n$, $n\ge n_0$ such that the resulting map h satisfies the desired properties. Let us first introduce an auxiliary family of functions $(f_p)_{p\in \mathbb {N}}$. For any p we define $f_p$ as the tent map $x\mapsto \max (x,1-x)$ on $[1/p,1/2-1/p]\cup [1/2+1/p,1-1/p]$. We extend it into a $C^r$ smooth interval map in such a way $f_p$ vanishes and admits a tangency of order r at the points 0, 1 / 2 and 1. Finally we extend $f_p$ periodically on the whole real axis. The intervals $[1/p,1/2-1/p]+k$ and $[1/2+1/p,1-1/p]+k$ for $k\in \mathbb {Z}$ are called the affine branches of $f_p$. Observe that the $C^r$ norm⁴ of $f_p$ may be chosen of order $p^{r}$. Then we let h be $x\mapsto \alpha _nf_{n^2}\left( (x-1+1/n) 2n^2N_n\right) +g(1-1/n)$ on $J_n$ where $\alpha _n\in \mathbb {R}^+$ and $N_n\in \mathbb {N}$ are chosen such that

• for each affine branch $I_n$ in $J_n$,

  $$\begin{aligned} h^{k}(I_n)\subset [0,1/\lambda ]\quad \text {for }k=1,\ldots ,r^n-1 \end{aligned}$$

  and

  $$\begin{aligned} h^{r^{n}}(I_n)=J_{n+1}, \end{aligned}$$

• the $C^r$ norm of h on $J_n$ goes to zero with n.

The first and second conditions are respectively fulfilled whenever

$$\begin{aligned} \lambda ^{r^{n}-1}\times \alpha _n(1/2-2/n^2)= 1/2(n+1)^2 \end{aligned}$$

and

$$\begin{aligned} \max _{k=1,\ldots ,r} \Vert f^{(k)}_{n^2}\Vert _\infty \times \alpha _n\times (2n^2N_n)^r\sim n^{2r}\times \alpha _n\times (2n^2N_n)^r=1/n. \end{aligned}$$

Fig. 2

The graph ofhon$J_n$in red. The arrows and intervals in blue represent the image $J_{n+1}$ of an affine branch $I_n$ under $h^{r^n}$

Conclusion Let $E_n=\bigcup _{I_n}I_n$ be the union of affine branches in $J_n$ and let $E=E_{n_0}\cap h^{-r^{n_0}}E_{n_0+1} \cap h^{-r^{n_0}-r^{n_0+1}}E_{n_0+2}\cap \cdots$ be the subset of points in $J_{n_0}$ visiting successively the sets $E_n$, $n\ge n_0$. Clearly E is contained in the basin of the Dirac measure at 0. To conclude it remains to see that E has positive Lebesgue measure and that $\chi (x)\ge \frac{\log \lambda }{r}$ for any x in E. The set E is an affine dynamically defined Cantor set where we remove a proportion of $4/n^2$ at the nth step. Therefore $\mathop {{\mathrm {Leb}}}(E)=\mathop {{\mathrm {Leb}}}(E_{n_0})\prod _{n>n_0}(1-4/n^2)>0$. Finally as $\log |h'|$ is equal on $I_n$ to $\log (\alpha _n4n^2N_n)\sim \frac{r-1}{r}\log \alpha _n\sim -r^{n-1}(r-1)\log \lambda$, the Lyapunov exponent at any $x\in E$ is given by

$$\begin{aligned} \chi (x)= & {} \limsup _p\frac{1}{p}\log |(h^p)'(x)|,\\= & {} \log \lambda \lim _q \frac{\sum _{q\ge n\ge n_0} \left( r^{n}-r^{n-1}(r-1)\right) }{\sum _{n\ge n_0} r^{n}},\\= & {} \frac{\log \lambda }{r}. \end{aligned}$$

Observe that any point in E is not recurrent.

Appendix B. Essential Range of $x\mapsto p\omega (x)$

We recall here the definition of the essential range of a Borel map with respect to a Borel measure. Finally we relate the set of physical-like measures of a topological system (M, f) with the essential range of $M\ni x\mapsto p\omega (x)$.

We consider two metric spaces X and Y with Y separable. Let m be a Borel measure on X and $\phi :X\rightarrow Y$ be a Borel map.

Definition 1

With the above notations the essential range $\overline{\mathop {{\mathrm {Im}}}}_m(\phi )$ of $\phi$ with respect to m is the complement of $\{y\in Y, \ \exists U \text { open with }y\in U \text { and }m(\phi ^{-1}U)=0\}$.

The set $\overline{\mathop {{\mathrm {Im}}}}_m(\phi )$ is a closed subset of Y and for m-almost every x the point $\phi (x)$ belongs to $\overline{\mathop {{\mathrm {Im}}}}_m(\phi )$. Moreover it is the smallest set satisfying these properties.

Lemma 8

Let (M, f) be a topological system. The map $p\omega : x\mapsto p\omega (x)$ from M to $\mathcal {KM}(M)$ is Borel.

Proof

As the set $\mathcal {KM}(M)$ is separable, it is enough to show $p\omega ^{-1}(B)$ is a Borel subset of M for any closed ball B of $\mathcal {KM}(M)$. Let B be the closed ball of radius $\epsilon$ centered at $K\in \mathcal {KM}(M)$, i.e. the set of compact subsets $K'$ of M with $K'\subset K_{\epsilon }$ and $K\subset K'_\epsilon$ where $K_\epsilon$ and $K'_\epsilon$ denote respectively the closed $\epsilon$-neighborhoods of K and $K'$. Firstly observe that $\{x\in M, \ p\omega (x)\subset K_{\epsilon }\}$ is closed. Then for a fixed sequence $(k_n)_{n\in \mathbb {N}}$ dense in K the following properties are equivalent:

$$\begin{aligned}&K\subset (p\omega (x))_\epsilon ,&\\ \Leftrightarrow&\mathfrak {d}(k_n, p\omega (x))\le \epsilon \qquad \qquad \text { for all }n,\\ \Leftrightarrow&\liminf _p \mathfrak {d} (k_n,\mu _x^p)<\epsilon ' \qquad \text { for all }n \text { and }\mathbb {Q}\ni \epsilon '>\epsilon . \end{aligned}$$

The functions $x\mapsto \mathfrak {d} (k_n,\mu _x^p)$ being continuous we conclude that $p\omega ^{-1}(B)$ is a Borel set. $\square$

Lemma 9

The set $\mathcal {PL}(m)$ of physical-like measures is the union of all $K\in \overline{\mathop {{\mathrm {Im}}}}_m(p\omega )$.

Proof

Firstly, the set $\overline{\mathop {{\mathrm {Im}}}}_m(p\omega )$ being a compact subset of $\mathcal {KM}(M)$, the set $\bigcup _{K\in \overline{\mathop {{\mathrm {Im}}}}_m(p\omega )}K$ is a compact subset of M. Therefore, from the definitions we get $\mathcal {PL}(m)\subset \bigcup _{K\in \overline{\mathop {{\mathrm {Im}}}}_m(p\omega )}K$. We argue by contradiction to prove the converse inclusion. Assume there is $K\in \overline{\mathop {{\mathrm {Im}}}}_m(p\omega )$ such that K is not contained in $\mathcal {PL}(m)$. Then this also holds for any $K'$ close enough to K. Therefore there exists an open neighborhood U of K such that $p\omega ^{-1}(U)$ has positive m-measure and for all x in this set $p\omega (x)$ is not contained in $\mathcal {PL}(m)$. It is impossible by definition of $\mathcal {PL}(m)$. $\quad \square$