Irregular sets for piecewise monotonic maps

TL;DR

This paper studies the irregular set in transitive piecewise monotonic maps, showing it is either empty or has full topological entropy, extending previous results and linking to the density of periodic measures.

Contribution

It generalizes Thompson's theorem to a broader class of maps and connects the irregular set properties to the Hofbauer-Raith problem.

Findings

Irregular set is either empty or has full topological entropy.

Generalizes Thompson's theorem for $eta$-transformations.

Links irregular set properties to the density of periodic measures.

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.