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Zero Lyapunov Exponents in Transitive Skew-products of Iterated Function Systems

Pablo G. Barrientos, Joel Angel Cisneros

Abstract

We study the class of transitive skew-products associated with iterated function systems of circle diffeomorphisms. We can approximate any transitive skew-product by maps in this class that have a robustly zero Lyapunov exponent. In particular, we prove the existence of non-hyperbolic ergodic measures for an open and dense subset of transitive skew-products. Moreover, these measures have full support and are the weak$^*$ limit of periodic measures.

Zero Lyapunov Exponents in Transitive Skew-products of Iterated Function Systems

Abstract

We study the class of transitive skew-products associated with iterated function systems of circle diffeomorphisms. We can approximate any transitive skew-product by maps in this class that have a robustly zero Lyapunov exponent. In particular, we prove the existence of non-hyperbolic ergodic measures for an open and dense subset of transitive skew-products. Moreover, these measures have full support and are the weak limit of periodic measures.
Paper Structure (6 sections, 11 theorems, 48 equations)

This paper contains 6 sections, 11 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['teo-B']}
  3. Preliminares
  4. Main Lemma
  5. Proof
  6. Proof of Theorem \ref{['thmAA']}

Key Result

Theorem A

For $k\geq 2$ and $1\leq r\leq \infty$, there exists an open and dense subset $\mathcal{R}$ of $\mathrm{TS}_{k}^{r}(\mathbb{S}^{1})$ such that for every $F\in\mathcal{R}$ there is an ergodic $F$-invariant probability measure $\mu$ with full support whose Lyapunov exponent $\lambda(\mu)$ equal to zer

Theorems & Definitions (19)

  • Theorem A
  • Theorem B
  • Remark 1.1
  • Remark 1.2
  • Proposition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Proposition 2.6
  • ...and 9 more