Irregular sets for a class of skew product transformations

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.