On the density of intermediate \beta-shifts of finite type

TL;DR

This paper characterizes the structure of intermediate eta-shifts of finite type, proving their density in the parameter space, thus extending classical results on greedy and lazy eta-shifts.

Contribution

It establishes the density of intermediate eta-shifts of finite type in the parameter space, generalizing Parry's 1960 classical results.

Findings

The set of intermediate eta-shifts of finite type is dense in the parameter space.

Generalization of Parry's classical results from 1960.

Provides a detailed structure of these eta-shifts.

Abstract

We determine the structure of the set of intermediate $\beta$-shifts of finite type. Specifically, we show that this set is dense in the parameter space $\Delta = \{ (\beta, \alpha) \in \mathbb{R}^{2} \colon \beta \in (1, 2) \; \text{and} \; 0 \leq \alpha \leq 2 - \beta\}$. This generalises the classical result of Parry from 1960 for greedy and (normalised) lazy $\beta$-shifts.