Induced Random $\beta$-transformation

TL;DR

This paper investigates the dynamics of the first return map in induced $eta$-transformations, characterizing parameter sets where the map exhibits generalized L"uroth series behavior and analyzing the structure of return time sequences.

Contribution

It provides a detailed analysis of the allowable return time sequences and identifies parameter sets where the first return map is a generalized L"uroth series transformation.

Findings

Existence of disjoint intervals where all return time sequences are permissible.

Identification of a Hausdorff dimension 1 set with measure zero where the map is a generalized L"uroth series.

Characterization of the parameter space for the induced $eta$-transformation.

Abstract

In this article we study the first return map defined on the switch region induced by the greedy and lazy maps. In particular we study the allowable sequences of return times, and when the first return map is a generalised L\"uroth series transformation. We show that there exists a countable collection of disjoint intervals $(\mathcal{I}_{n})_{n=1}^{\infty},$ such that all sequences of return times are permissible if and only if $\beta\in \mathcal{I}_{n}$ for some $n$. Moreover, we show that there exists a set $M\subseteq(1,2)$ of Hausdorff dimension $1$ and Lebesgue measure zero, for which the first return map is a generalised L\"uroth series transformation if and only if $\beta\in M$.