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- Title
On the specification property and synchronization of unique q -expansions.
- Authors
BARRERA, RAFAEL ALCARAZ
- Abstract
Given a positive integer M and $q \in (1, M+1]$ we consider expansions in base q for real numbers $x \in [0, {M}/{q-1}]$ over the alphabet $\{0, \ldots , M\}$. In particular, we study some dynamical properties of the natural occurring subshift $(\boldsymbol{{V}}_q, \sigma)$ related to unique expansions in such base q. We characterize the set of $q \in \mathcal {V} \subset (1,M+1]$ such that $(\boldsymbol{{V}}_q, \sigma)$ has the specification property and the set of $q \in \mathcal {V}$ such that $(\boldsymbol{{V}}_q, \sigma)$ is a synchronized 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 as subsets of $\mathcal {V}$ giving similar results to those shown by Blanchard [ 10 ] and Schmeling in [ 36 ] in the context of $\beta $ -transformations.
- Publication
Ergodic Theory & Dynamical Systems, 2021, Vol 41, Issue 9, p2659
- ISSN
0143-3857
- Publication type
Article
- DOI
10.1017/etds.2020.55