Measures of Intermediate Entropies for Skew Product Diffeomorphisms

TL;DR

This paper investigates skew product maps with positive entropy, demonstrating the existence of ergodic measures with intermediate entropies by constructing specific return sets with horseshoe structures.

Contribution

It introduces a method to construct ergodic measures of intermediate entropy for skew product diffeomorphisms with positive entropy and nonzero Lyapunov exponents.

Findings

Existence of ergodic measures with intermediate entropy.

Construction of return sets with horseshoes along fibers.

Maximum entropy of these measures can approach the system's total entropy.

Abstract

In this paper we study a skew product map $F$ with a measure $\mu$ of positive entropy. We show that if on the fibers the map are $C^{1+\alpha}$ diffeomorphisms with nonzero Lyapunov exponents, then $F$ has ergodic measures of intermediate entropies. To construct these measures we find a set on which the return map is a skew product with horseshoes along fibers. We can control the average return time and show the maximum entropy of these measures can be arbitrarily close to $h_\mu(F)$.