Nonhyperbolic step skew-products: Ergodic approximation

TL;DR

This paper investigates nonhyperbolic skew-product systems with intermingled contracting and expanding regions, showing that zero-exponent ergodic measures can be approximated by hyperbolic measures and can be perturbed between positive and negative exponents, with entropy considerations.

Contribution

It introduces axioms capturing key dynamics of nonhyperbolic transitive maps and proves approximation and perturbation results for ergodic measures with zero fiber exponent.

Findings

Zero fiber exponent measures are approximated by hyperbolic measures in weak* topology and entropy.

Measures with negative exponent can be perturbed to positive exponent measures, losing entropy.

The entropy loss is proportional to the difference in Lyapunov exponents.

Abstract

We study transitive step skew-product maps modeled over a complete shift of $k$, $k\ge2$, symbols whose fiber maps are defined on the circle and have intermingled contracting and expanding regions. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents. We introduce a set of axioms for the fiber maps and study the dynamics of the resulting skew-product. These axioms turn out to capture the key mechanisms of the dynamics of nonhyperbolic robustly transitive maps with compact central leaves. Focusing on the nonhyperbolic ergodic measures (with zero fiber exponent) of these systems, we prove that such measures are approximated in the weak$\ast$ topology and in entropy by hyperbolic ones. We also prove that they are in the intersection of the convex hulls of the measures with positive fiber exponent and with negative fiber exponent. Our methods also allow us to perturb hyperbolic measures. We can perturb a measure with negative exponent directly to a measure with positive exponent (and vice-versa), however we lose some amount of entropy in this process. The loss of entropy is determined by the difference between the Lyapunov exponents of the measures.