# On the specification property and synchronisation of unique $q$-expansions

## TL;DR

This paper investigates the dynamical properties of subshifts related to unique base-$q$ expansions, characterizing when they exhibit the specification property or are synchronized, based on the combinatorial structure of quasi-greedy expansions.

## Contribution

It provides a complete characterization of the sets of $q$ for which the subshift has the specification property or is synchronized, extending the understanding of unique $q$-expansions.

## Key findings

- Characterizes $q$ with the specification property.
- Identifies $q$ for synchronized subshifts.
- Determines the size of these classes.

## Abstract

Given a positive integer $M$ and $q \in (1, M+1]$ we consider expansions in base $q$ for real numbers $x \in \left[0, {M}/{q-1}\right]$ over the alphabet $\{0, \ldots, M\}$. In particular, we study some dynamical properties of the natural occurring subshift $(\mathbf{V}_q, \sigma)$ related to unique expansions in such base $q$. We characterise the set of $q \in (1,M+1]$ such that $(\mathbf{V}_q, \sigma)$ has the specification property and the set of $q \in (1,M+1]$ such that $(\mathbf{V}_q, \sigma)$ is a synchronised subshift. Such properties are studied by analysing the combinatorial and dynamical properties of the quasi-greedy expansion of $q$. We also calculate the size of such classes giving similar results to those shown by Schmeling in (Ergodic Theory and Dynamical Systems, 17:675--694, 6 1997) in the context of $\beta$-transformations.