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$\times a$ and $\times b$ empirical measures, the irregular set and entropy

Part of: Diophantine approximation, transcendental number theory Ergodic theory Low-dimensional dynamical systems

Published online by Cambridge University Press:  15 August 2023

SHUNSUKE USUKI
SHUNSUKE USUKI*
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kitashirakawa Oiwake-cho, Kyoto 606-8502, Japan
*
e-mail: usuki.shunsuke.72r@st.kyoto-u.ac.jp
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Abstract

For integers a and $b\geq 2$, let $T_a$ and $T_b$ be multiplication by a and b on $\mathbb {T}=\mathbb {R}/\mathbb {Z}$. The action on $\mathbb {T}$ by $T_a$ and $T_b$ is called $\times a,\times b$ action and it is known that, if a and b are multiplicatively independent, then the only $\times a,\times b$ invariant and ergodic measure with positive entropy of $T_a$ or $T_b$ is the Lebesgue measure. However, it is not known whether there exists a non-trivial $\times a,\times b$ invariant and ergodic measure. In this paper, we study the empirical measures of $x\in \mathbb {T}$ with respect to the $\times a,\times b$ action and show that the set of x such that the empirical measures of x do not converge to any measure has Hausdorff dimension one and the set of x such that the empirical measures can approach a non-trivial $\times a,\times b$ invariant measure has Hausdorff dimension zero. Furthermore, we obtain some equidistribution result about the $\times a,\times b$ orbit of x in the complement of a set of Hausdorff dimension zero.

Keywords

multiplication on the circleirregular setentropyHausdorff dimensionequidistribution

MSC classification

Primary: 37E10: Maps of the circle 37A35: Entropy and other invariants, isomorphism, classification
Secondary: 11J71: Distribution modulo one
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 20
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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