The dynamics of expansive invertible onesided cellular automata

Abstract

Abstract. Using textile systems, we prove the conjecture of Boyle and Maass that the dynamical system defined by an expansive invertible onesided cellular automaton is topologically conjugate to a topological Markov shift. We also study expansive leftmost-permutive onesided cellular automata and bipermutive endomorphisms of mixing topological Markov shifts.

Textile systems and one-sided resolving automorphisms and endomorphisms of the shift

“…This corollary is a generalization of the main theorem of [23] that an expansive automorphism of a mixing one-sided SFT is conjugate to an SFT. Blanchard and Tisseur [5] proved that if an onto endomorphism ϕ of a topologicallytransitive SFT (X, σ ) has a point of equicontinuity, then ϕ has periodic points dense.…”

“…However, from Fiebig's result until Boyle's result, no progress on the problem was made. This contrasts with the complete solution of the similar problems for positively-expansive endomorphisms of one-sided and two-sided SFTs and expansive automorphisms of one-sided SFTs and remarkable further progress on these and related subjects [4,7,8,12,13,16,[20][21][22][23]25] (for a quick survey, see [24]). The open problem above is extended to the following problem: for any automorphism ϕ of a transitive SFT (X, σ ), determine the dynamics of ϕ i σ j for i, j ∈ Z.…”

“…Before proving the above results, we prove two main results on textile systems: Propositions 3.10 and 4.3. The former is proved by using a method developed in [23] and a fundamental result of Boyle in [6]; the latter is proved by using the method of [23] with a new ingredient, the chain recurrence condition, which in symbolic dynamics was explained by Boyle [6]. Our result stated first is proved in two ways: one uses Proposition 3.10 and the other uses Proposition 4.3, and the generalized result stated next is proved by using Proposition 4.3.…”

On Dynamical Complexity of Surjective Ultimately Right-Expansive Cellular Automata

“…Questions regarding what kind of a subshift an expansive cellular automaton is conjugate to has been considered at least in [16,10,4,2,17,18]. In [16] Nasu presented his textile systems and used them to show that positively expansive two-sided cellular automata are conjugate to full shifts.…”

“…Blanchard and Maass [4] improved the case of positively expansive one-sided cellular automata and showed that these are conjugate to full shifts. Next Nasu [17], again using textile systems, proved that one-sided expansive cellular automata are conjugate to subshifts of finite type. Later on Nasu [18] proved that also expansive two-sided cellular automata are conjugate to subshifts of finite type if the local neighborhood is one-sided.…”