Bijective and general arithmetic codings for Pisot automorphisms of the torus
| dc.creator | Sidorov, Nikita | |
| dc.date | 2000-06-21 | |
| dc.date.accessioned | 2026-07-07T04:36:00Z | |
| dc.date.available | 2026-07-07T04:36:00Z | |
| dc.description | Let $T$ be an algebraic automorphism of $\mathbb{T}^{m}$ having the following property: the characteristic polynomial of its matrix is irreducible over $\mathbb{Q}$, and a Pisot number $β$ is one of its roots. We define the mapping $ϕ_{\mathbf{t}}$ acting from the two-sided $β$-compactum onto $\mathbb{T}^{m}$ as follows: \[ ϕ_{\mathbf{t}}(\barε)= \sum_{k\in\mathbb{Z}}ε_{k}T^{-k}\mathbf{t}, \] where $\mathbf{t}$ is a fundamental homoclinic point for $T$, i.e., a point homoclinic to $\mathbf{0}$ such that the linear span of its orbit is the whole homoclinic group (provided such a point exists). We call such a mapping an arithmetic coding of $T$. This paper is aimed to show that under some natural hypothesis on $β$ (which is apparently satisfied for all Pisot units) the mapping $ϕ_{\mathbf{t}}$ is bijective a.e. with respect to the Haar measure on the torus. Besides, we study the case of more general parameters $\mathbf{t}$, not necessarily fundamental, and relate the number of preimages of $ϕ_{\mathbf{t}}$ to certain number-theoretic quantities. We also give several full criteria for $T$ to admit a bijective arithmetic coding. This work continues the study begun in [Sidorov & Vershik, Journal Dynam. Control Systems 4 (1998), 365-399] for the special case $m=2$. | |
| dc.description | 25 pages, Latex | |
| dc.identifier | https://arxiv.org/abs/math/0006159 | |
| dc.identifier | http://arxiv.org/abs/math/0006159 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/59448 | |
| dc.subject | Dynamical Systems | |
| dc.subject | Number Theory | |
| dc.subject | 28D05, 37C29; 11R06, 11R04 | |
| dc.title | Bijective and general arithmetic codings for Pisot automorphisms of the torus | |
| dc.type | text |