A continuous action of a group G on a compact metric space has sensitive dependence on initial conditions if there is a number ε > 0 such that for any open set U we can find g ∈ G such that g.U has diameter greater than ε. We prove that if a G action preserves a probability measure of full support, then the system is either minimal and equicontinuous, or has sensitive dependence on initial cond...

Revisiting the notion of μ-almost equicontinuous cellular automata introduced by R. Gilman, we show that the sequence of image measures of a shift ergodic measure μ by iterations of a μ-almost equicontinuous cellular automata F , converges in Cesaro mean to an invariant measure μc. If the initial measure μ is a Bernouilli measure, we prove that the Cesaro mean limit measure μc is shift mixing. ...

For an infinite discrete group $ G acting on a compact metric space X $, we introduce several weak versions of equicontinuity along subsets and show that if minimal system (X, G) admits invariant measure then is distal only it pairwise IP$ ^* $-equicontinuous; the product (X\times X, has dense set points, $-equicontinuous central$ with being abelian, order \infty FIP$ $-equicontinuous.

In this paper it is proved that if a minimal system has the property its sequence entropy uniformly bounded for all sequences, then only finitely many ergodic measures and an almost finite to one extension of maximal equicontinuous factor. This result obtained as application general criterion which states under amenable group action factor no infinite independent sets length k some k≥2, measures.

Revisiting the notion of μ-almost equicontinuous cellular automata introduced by R. Gilman, we show that the sequence of image measures of a shift ergodic measure μ by iterations of such automata converges in Cesaro mean to an invariant measure μc. If the initial measure μ is a Bernouilli measure, we prove that the Cesaro mean limit measure μc is shift mixing. Therefore we also show that for an...

The generic limit set of a dynamical system is the smallest that attracts most space in topological sense: it closed with comeager basin attraction. Introduced by Milnor, has been studied context one-dimensional cellular automata Djenaoui and Guillon, Delacourt, Törmä. In this article we present complexity bounds on realizations sets prescribed properties. We show have Π20 language if they are ...

1. (a) Let (X, ‖ · ‖) and (Y, ‖ · ‖) be normed spaces. Show that the following conditions on a family T ⊆ L(X, Y ) are logically equivalent: (1) {‖T ‖ : T ∈ T} is bounded in R; (2) {T : T ∈ T} is equicontinuous at some point of X ; (3) {T : T ∈ T} is equicontinuous at 0 ∈ X ; (4) {T : T ∈ T} is equicontinuous at every point of X . (This order may not be the most efficient one to prove round-rob...

A new class of dynamical systems is defined, the class of “locally equicontinuous systems” (LE). We show that the property LE is inherited by factors as well as subsystems, and is closed under the operations of pointed products and inverse limits. In other words, the locally equicontinuous functions in l∞(Z) form a uniformly closed translation invariant subalgebra. We show that WAP ⊂ LE ⊂ AE, w...

1 Glossary Almost equicontinuous CA has an equicontinuous configuration. Attractor: omega-limit of a clopen invariant set. Blocking word interrupts information flow. Closing CA: distinct asymptotic configurations have distinct images. Column subshift: columns in space-time diagrams. Cross section: one-sided inverse map. ∗Université de Nice Sophia Antipolis, Département d’Informatique, Parc Valr...