Intermediate $\beta$-shifts as greedy $\beta$-shifts with a hole

TL;DR

This paper characterizes intermediate β-transformations as conjugate to greedy β-transformations with a hole at zero, leading to new results on survivor sets, Hausdorff dimension, and badly approximable numbers in non-integer bases.

Contribution

It establishes a topological conjugacy between intermediate and greedy β-transformations with a hole, and extends metric number theory results on survivor sets and badly approximable numbers.

Findings

Topological conjugacy between intermediate and greedy β-transformations with a hole at zero.

New methods to calculate Hausdorff dimension of survivor sets.

Sets of badly approximable numbers are winning in Schmidt games under certain conditions.

Abstract

We show that every intermediate $\beta$-transformation is topologically conjugate to a greedy $\beta$-transformation with a hole at zero, and provide a counterexample illustrating that the correspondence is not one-to-one. This characterisation is employed to (1) build a Krieger embedding theorem for intermediate $\beta$-transformation, complementing the result of Li, Sahlsten, Samuel and Steiner [2019], and (2) obtain new metric and topological results on survivor sets of intermediate $\beta$-transformations with a hole at zero, extending the work of Kalle, Kong, Langeveld and Li [2020]. Further, we derive a method to calculate the Hausdorff dimension of such survivor sets as well as results on certain bifurcation sets. Moreover, by taking unions of survivor sets of intermediate $\beta$-transformations one obtains an important class of sets arising in metric number theory, namely sets of badly approximable numbers in non-integer bases. We prove, under the assumption that the underlying symbolic space is of finite type, that these sets of badly approximable numbers are winning in the sense of Schmidt games, and hence have the countable intersection property, extending the results of Hu and Yu [2014], Tseng [2009] and F\"{a}rm, Persson and Schmeling [2010].