Irregular sets for a class of skew product transformations

TL;DR

This paper investigates the properties of irregular sets in skew product transformations driven by uniquely ergodic systems, showing they are often residual and carry full entropy under certain conditions.

Contribution

It establishes the residuality and entropy fullness of irregular sets for a class of skew product transformations with fiber specification.

Findings

Irregular sets are nonempty and residual on fibers.

Irregular sets carry full fiber topological entropy.

Results depend on nonemptiness of irregular sets on individual fibers.

Abstract

In this paper, we study the irregular set of any continuous observable for a class of skew product transformations, which is driven by a uniquely ergodic homeomorphism system $(\Omega,\mathbb{P},\theta)$ and satisfies Anosov and toplogical mixing on fibers property. We prove that if the irregular set of any continuous observable is nonempty on a fiber, then the irregular set must be nonempty, residual, and carry full fiber topological entropy on $\mathbb{P}$-a.e. fibers. The orbit-gluing technique provided by the fiber specification property are utilized.