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2021 Irregular Sets for Piecewise Monotonic Maps
Yushi NAKANO, Kenichiro YAMAMOTO
Tokyo J. Math. Advance Publication 1-12 (2021). DOI: 10.3836/tjm/1502179349

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 the Hofbauer-Raith problem on the density of periodic measures.

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Yushi NAKANO. Kenichiro YAMAMOTO. "Irregular Sets for Piecewise Monotonic Maps." Tokyo J. Math. Advance Publication 1 - 12, 2021. https://doi.org/10.3836/tjm/1502179349

Information

Published: 2021
First available in Project Euclid: 4 October 2021

Digital Object Identifier: 10.3836/tjm/1502179349

Subjects:
Primary: 37B10
Secondary: 37B40, 37E05

Rights: Copyright © 2021 Publication Committee for the Tokyo Journal of Mathematics

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