Suppose P ⊆ N and let (ΣP , σP ) be the spacing shift defined by P . We show that if the topological entropy h(σP ) of a spacing shift is equal zero, then (ΣP , σP ) is proximal. Also h(σP ) = 0 if and only if P = N \ E where E is an intersective set. Moreover, we show that h(σP ) > 0 implies that P is a ∆ ∗-set; and by giving a class of examples, we show that this is not a sufficient condition...

Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy vanishes whenever the repetitivity function satisfies a certain growth restriction. 2000 AMS Subject Classification: 37A35, 37B40, 52C23

Let X be an uncountable compact metrizable space of topological dimension zero. Given any a ∈ [0,∞] there is a homeomorphism on X whose topological entropy is a.

The repetition property of a dynamical system, a notion introduced in [2], plays an importance role in analyzing spectral properties of ergodic Schrödinger operators. In this paper, entropy of dynamical systems with repetition property is investigated. It is shown that the topological entropy of dynamical systems with the global repetition property is zero. Minimal dynamical systems having both...

For a compact Hausdorff space X, let J be the ordered set associated with of all finite open covers X such that there exists nJ, where nJ is dimension ∂. Therefore, we have Hˇp(X;Z), 0≤p≤n=nJ. continuous self-map f on α∈J an cover and Lf(α)={Lf(U)|U∈α}. Then, fiber L˙f(α) Xf induced by Lf(α). In this paper, define topological entropy entL(f) as supremum ent(f,L˙f(α)) through Xf={Lf(U);U⊂X}, Lf(...

The purpose of this paper is to present a weighted kneading theory for one-dimensional maps with a hole. We consider extensions of the kneading theory of Milnor and Thurston to expanding discontinuous maps with a hole and introduce weights in the formal power series. This method allows us to derive techniques to compute explicitly the topological entropy, the Hausdorff dimension, and the escape...

First we show that any group of automorphisms of null-entropy of a projective hyperkähler manifold M is almost abelian of rank at most ρ(M) − 2. We then characterize automorphisms of a K3 surface with nullentropy and those with positive entropy in algebro-geometric terms. We also give an example of a group of automorphisms which is not almost abelian in each dimension.

Our automatic method developed for the detection of chaos is used for finding rigorous lower bounds for the topological entropy of the classical Hénon mapping. We do this within the abstract framework created by Galias and Zgliczynski in 2001, and focus on covering graphs involving different iterations. Our results are compared to those obtained by them.