1 results
$\times a$ and 
$\times b$ empirical measures, the irregular set and entropy
- Part of
-
- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 6 / June 2024
- Published online by Cambridge University Press:
- 15 August 2023, pp. 1673-1692
- Print publication:
- June 2024
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- Article
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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.
