Variational Principle for Nonhyperbolic Ergodic Measures: Skew Products and Elliptic Cocycles

Abstract

For a large class of transitive non-hyperbolic systems, we construct nonhyperbolic ergodic measures with entropy arbitrarily close to its maximal possible value. The systems we consider are partially hyperbolic with one-dimensional central direction for which there are positive entropy ergodic measures whose central Lyapunov exponent is negative, zero, or positive. We construct ergodic measures with zero central Lyapunov exponent whose entropy is positive and arbitrarily close to the topological entropy of the set of points with central Lyapunov exponent zero. This provides a restricted variational principle for nonhyperbolic (zero exponent) ergodic measures. The result is applied to the setting of $\mathrm {SL}(2,\mathbb {R})$ matrix cocycles and provides a counterpart to Furstenberg’s classical result: for an open and dense subset of elliptic $\mathrm {SL}(2,\mathbb {R})$ cocycles we construct ergodic measures with upper Lyapunov exponent zero and with metric entropy arbitrarily close to the topological entropy of the set of infinite matrix products with subexponential growth of the norm.

Appendix: Proof of Proposition 11.1

Given $\phi :X\rightarrow \mathbb {R}$, denote

$$\begin{aligned} \underline{\phi }(x)\mathop {=}\limits ^{\mathrm{def}}\liminf _{n\rightarrow \infty }\frac{1}{n}\sum _{k=0}^{n-1}\phi (G^k(x)) {\quad \text { and }\quad } \overline{\phi }(x)\mathop {=}\limits ^{\mathrm{def}}\limsup _{n\rightarrow \infty }\frac{1}{n}\sum _{k=0}^{n-1}\phi (G^k(x)). \end{aligned}$$

Given $\varepsilon >0$, denote the upper topological limit of $(\Upsilon _{n,\phi ,\varepsilon })_n$ by $\Upsilon _{\phi ,\varepsilon }$, that is,

$$\begin{aligned} \Upsilon _{\phi ,\varepsilon } \mathop {=}\limits ^{\mathrm{def}}{\bigcap _{k=1}^\infty \overline{\bigcup _{n=k}^\infty \Upsilon _{n,\phi ,\varepsilon }}} =\big \{y\in X:\exists n_k\rightarrow \infty , y_k\in \Upsilon _{n_k,\phi ,\varepsilon },y=\lim _{k\rightarrow \infty }y_k\big \}. \end{aligned}$$

We use the following fact that is straightforward to check.

Claim 11.2

For every continuous $\phi :X\rightarrow \mathbb {R}$ and $\varepsilon >0$ it holds

$$\begin{aligned} \varrho (\Upsilon _{\phi ,\varepsilon })\ge \limsup _{n\rightarrow \infty }\varrho _n(\Upsilon _{n,\phi ,\varepsilon }). \end{aligned}$$

Lemma 11.3

For every continuous $\phi :X\rightarrow \mathbb {R}$ and $\varepsilon >0$ there exists a set $\Xi _{\phi ,\varepsilon }\subset \Upsilon _{\phi ,\varepsilon }$ such that $\varrho (\Xi _{\phi ,\varepsilon })>1-\varepsilon$ and

$$\begin{aligned} \int \phi \,d\varrho -\varepsilon<\overline{\phi }(x)=\underline{\phi }(x) <\int \phi \,d\varrho +\varepsilon \end{aligned}$$

for every $x\in \Xi _{\phi ,\varepsilon }$.

Proof

Given $\phi$ and $\varepsilon$, let $L=L(\phi ,\varepsilon )$ and for $\ell \ge L$ let $N=N(\ell )$ be as in the hypothesis of the proposition. By Claim 11.2 and our hypothesis,

$$\begin{aligned} \varrho (\Upsilon _{\phi ,\varepsilon })\ge \limsup _n\varrho _n(\Upsilon _{n,\phi ,\varepsilon })>1-\varepsilon . \end{aligned}$$

Every $x\in \Upsilon _{\phi ,\varepsilon }$ is the limit of some sequence of points $x_i$ in $\Upsilon _{n_i,\phi ,\varepsilon }$, $n_i\ge \ell$. Hence, for $n_i\ge \ell$ sufficiently large, it holds

$$\begin{aligned} \left|\frac{1}{T(\ell )}\sum _{k=0}^{T(\ell )-1}\phi (G^k(x)) -\frac{1}{T(\ell )}\sum _{k=0}^{T(\ell )-1}\phi (G^k(x_i))\right|<\varepsilon . \end{aligned}$$

By our hypothesis on $x_i\in \Upsilon _{n_i,\phi ,\varepsilon }$, it holds

$$\begin{aligned} \left|\frac{1}{T(\ell )}\sum _{k=0}^{T(\ell )-1}\phi (G^k(x_i)) - \int \phi \,d\varrho \right|<\varepsilon . \end{aligned}$$

Hence, for every $x\in \Upsilon _{\phi ,\varepsilon }$ and $\ell \ge 1$ sufficiently large it holds

$$\begin{aligned} \left|\frac{1}{T(\ell )}\sum _{k=0}^{T(\ell )-1}\phi (G^k(x)) - \int \phi \,d\varrho \right|<2\varepsilon . \end{aligned}$$

Therefore, with the notation above, for every $x\in \Upsilon _{\phi ,\varepsilon }$

$$\begin{aligned} \overline{\phi }(x) > \int \phi \,d\varrho -2\varepsilon {\quad \text { and }\quad } \underline{\phi }(x) < \int \phi \,d\varrho +2\varepsilon . \end{aligned}$$

Applying the Birkhoff theorem to the invariant measure $\varrho$, we get a set Z with $\varrho (Z)=1$ so that at every $z\in Z$ it holds $\overline{\phi }(z)=\underline{\phi }(z)$. By the above, for every $z\in \Xi _{\phi ,\varepsilon }\mathop {=}\limits ^{\mathrm{def}}Z\cap \Upsilon _{\phi ,\varepsilon }$ it holds $|\overline{\phi }(z)-\phi (\varrho )|<3\varepsilon$ and $\varrho (\Xi _{\phi ,\varepsilon })=\varrho (\Upsilon _{\phi ,\varepsilon })>1-\varepsilon$. This proves the lemma.

$\square$

Let us now prove that $\varrho$ is ergodic. Take a dense set of continuous functions $\{\phi _k\}_k$ and a summable sequence of positive numbers $(\varepsilon _k)_k$. As

$$\begin{aligned} \sum _k\varrho (\Upsilon _{\phi _k,\varepsilon _k}^c) \le \sum _k\varepsilon _k<\infty , \end{aligned}$$

by the Borel–Cantelli lemma, there is a set $\Upsilon$ satisfying $\varrho (\Upsilon )=1$ such that every $x\in \Upsilon$ is contained in only finitely many sets $\Upsilon _{\phi _k,\varepsilon _k}^c$. It follows that for every continuous $\phi$ and $x\in \Upsilon$ Birkhoff averages of $\phi$ converge to $\int \phi \,d\varrho$. This implies that $\varrho$ is G-ergodic.

$\square$