Diophantine approximation of the orbit of 1 in the dynamical system of bete expansions

TL;DR

This paper investigates the Hausdorff dimension of the set of parameters eta>1 for which the orbit of 1 under eta-transformations approximates a given point with a specified precision, revealing intricate distribution properties.

Contribution

It introduces the recurrence time of words in symbolic space to analyze the distribution and size of cylinders in the eta parameter space, advancing understanding of eta-expansion dynamics.

Findings

Determines the Hausdorff dimension of approximation sets for eta-transformations.

Introduces recurrence time of words to characterize cylinder distributions.

Provides a measure of how well points can be approximated by orbits in eta systems.

Abstract

We consider the distribution of the orbits of the number 1 under the $\beta$-transformations $T_\beta$ as $\beta$ varies. Mainly, the size of the set of $\beta>1$ for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. That is, the dimension of the following set $$ E\big({\ell_n}_{n\ge 1}, x_0\big)=\Big{\beta>1: |T^n_{\beta}1-x_0|<\beta^{-\ell_n}, {for infinitely many} n\in \N\Big} $$ is determined, where $x_0$ is a given point in $[0,1]$ and ${\ell_n}_{n\ge 1}$ is a sequence of integers tending to infinity as $n\to \infty$. For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterize the lengths and the distribution of cylinders (the set of $\beta$ with a common prefix in the expansion of 1) in the parameter space ${\beta\in \R: \beta>1}$.