A Dichotomy for the dimension of SRB measure

TL;DR

This paper establishes a dichotomy for the SRB measure in skew product dynamical systems, showing it is either supported on an analytic graph or has a specific fractal dimension, with the latter being typical unless the system is trivial.

Contribution

It proves a dichotomy for the SRB measure's support and dimension in skew product systems, revealing when the measure is supported on an analytic graph versus having a fractal dimension.

Findings

Support of SRB measure is either an analytic graph or has a specific fractal dimension.

The analytic graph support occurs only for finitely many parameters unless the function is constant.

The dimension of the SRB measure is explicitly given by a formula involving system parameters.

Abstract

We study dynamical systems generated by skew products: $$T: [0,1)\times\mathbb{R}\to [0,1)\times\mathbb{R} \quad\quad T(x,y)=(bx\mod1,\gamma y+\phi(x))$$ where integer $b\ge2$, $0<\gamma<1$ and $\phi$ is a real analytic $\mathbb{Z}$-periodic function. We prove the following dichotomy for the SRB measure $\omega$ for $T$: Either the support of $\omega$ is a graph of real analytic function, or the dimension of $\omega$ is equal to $\min\{2,1+\frac{\log b}{\log1/\gamma}\}$. Furthermore, given $b$ and $\phi$, the former alternative only happens for finitely many $\gamma$ unless $\phi$ is constant.