xoloha.blogg.se - Subshift disjoint from a given subshift

We call alphabet a ﬁnite set of elements called letters. In the last section we consider the case of substitution subshiftsand we give a necessary and suﬃcient condition for such subshifts to havea ﬁnite number (up to isomorphism) of Cantor factors and to have a ﬁnitenumber (up to isomorphism) of non-periodic Cantor factors. We take the opportunity to make some comments on the result werecalled before concerning chains of factors, we show that LR subshifts arecoalescent and that if two LR subshifts are weakly isomorphic then they areisomorphic. Section 5 is devoted to the proof of Theo-rem 1. In Section 4 we established that LRsubshifts are uniquely ergodic. This last result was certainly knownbut the author did not ﬁnd reference. they do not have proper Cantor factor, unlessit is well-known they are not prime. We also give a necessary and suﬃcient condi-tion for a sturmian subshift to be LR, and we prove that sturmian subshifts( X, T ) are Cantor prime, i.e. In Section 2 we recall these properties.In Section 3 we characterize LR subshifts by means of S -adic subshifts, thisnotion was introduced in. The proof of this result intensively uses the notion of return words and theirproperties established in. The set of its non-periodic subshift factors is ﬁnite up to isomorphism. Let ( X, T ) be a linearly recurrent subshift. Anatural question arise: Do LR subshifts have a ﬁnite number of subshiftfactors (up to isomorphism) ? The main result of this paper answers to thisquestion: Theorem 1

γ −→ ( Y, T ) γ −→ ( Y, T ), n ≥ D then there exists an integer i for which γ i is an isomorphism.

The present work is motivatedby the fact, proved in, that given a LR subshift ( X, T ) there exists aconstant D such that for all chains of non-periodic subshift factors( X, T ) γ n − −→ ( Y n −, T ) γ n − −→

$subshift disjoint from a given subshift$

We focus ourattention on their topological Cantor factors. X, T ) is linearly recurrent if there existsa constant K so that for each clopen set U generated by a ﬁnite word u the return time to U, with respect to T, is bounded by K | u |.

In this paper we continue the study of linearly recurrent (LR) subshifts initi-ated in. We also give a constructive characterization ofthese subshifts. We prove that givena linearly recurrent subshift ( X, T ) the set of its non-periodic subshift factorsis ﬁnite up to isomorphism. X, T ) is linearly recurrent if there existsa constant K so that for each clopen set U generated by a ﬁnite word u thereturn time to U, with respect to T, is bounded by K | u |. J u l Linearly recurrent subshifts have aﬁnite number of non-periodic subshiftfactors