Nonwandering sets of interval skew products

TL;DR

This paper studies skew product systems over subshifts with interval fibers, showing that for most parameters, the nonwandering set has zero measure, indicating typical dynamical behavior with minimal recurrent points.

Contribution

It proves that in a broad class of skew products, the nonwandering set is measure-zero for almost all parameters, extending understanding of typical dynamics in these systems.

Findings

Nonwandering set has zero measure for almost all parameters.

Residual subset of skew products also exhibits zero measure nonwandering sets.

Results apply to a natural class of 1-parameter families over subshifts.

Abstract

In this paper we consider a class of skew products over transitive subshifts of finite type with interval fibers. For a natural class of 1-parameter families we prove that for all but countably many parameter values the nonwandering set (in particular, the union of all attractors and repellers) has zero measure. As a consequence, the same holds for a residual subset of the space of skew products.