Irregular sets for piecewise monotonic maps

Abstract

For any transitive piecewise monotonic map for which the set of periodic measures is dense in the set of ergodic invariant measures (such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces), we show that the set of points for which the Birkhoff average of a continuous function does not exist (called the irregular set) is either empty or has full topological entropy. This generalizes Thompson's theorem for irregular sets of $\beta$-transformations, and reduces a complete description of irregular sets of transitive piecewise monotonic maps to Hofbauer-Raith problem on the density of periodic measures.

11 pages; strengthened the main result (Theorem 1) by introducing Hofbauer's topological entropy for piecewise monotonic maps

References in corpus (5)

• Takens' last problem and existence of non-trivial wandering domains
• Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets
• Geometric Lorenz flows with historic behavior
• Multifractal analysis of the irregular set for almost-additive sequences via large deviations
• Abundance of wild historic behavior

Cited by in corpus (1)

• Topological entropy and Hausdorff dimension of irregular sets for non-hyperbolic dynamical systems