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 the Anosov and topological mixing on fibers properties. The orbit-gluing technique provided by the fiber specification property is utilized. For such systems, we prove the following rigidity phenomenon 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. This result can be applied to a class of partially hyperbolic systems.
Fields of Study & Subjects
Mathematical Dynamics and Fractals
Cellular Automata and Applications
Advanced Differential Equations and Dynamical Systems
Publication details
- Journal / source
- Discrete and Continuous Dynamical Systems
- Publisher
- American Institute of Mathematical Sciences (AIMS)
- Volume
- 51
- Issue
- 0
- Pages
- 128–160
- Publication date
- 1 January 2026