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Topological size of scrambled sets

François Blanchard, Wen Huang, L'ubomír Snoha (2008)

Colloquium Mathematicae

A subset S of a topological dynamical system (X,f) containing at least two points is called a scrambled set if for any x,y ∈ S with x ≠ y one has l i m i n f n → ∞ d ( f ⁿ ( x ) , f ⁿ ( y ) ) = 0 and l i m s u p n → ∞ d ( f ⁿ ( x ) , f ⁿ ( y ) ) > 0 , d being the metric on X. The system (X,f) is called Li-Yorke chaotic if it has an uncountable scrambled set. These notions were developed in the context of interval maps, in which the existence of a two-point scrambled set implies Li-Yorke chaos and many other chaotic properties. In the present paper we address several questions about scrambled...

Topological transitivity of solvable group actions on the line ℝ

Suhua Wang, Enhui Shi, Lizhen Zhou, Grant Cairns (2009)

Colloquium Mathematicae

Let ϕ:G → Homeo₊(ℝ) be an orientation preserving action of a discrete solvable group G on ℝ. In this paper, the topological transitivity of ϕ is investigated. In particular, the relations between the dynamical complexity of G and the algebraic structure of G are considered.

Topology and dynamics of unstable attractors

M. A. Morón, J. J. Sánchez-Gabites, J. M. R. Sanjurjo (2007)

Fundamenta Mathematicae

This article aims to explore the theory of unstable attractors with topological tools. A short topological analysis of the isolating blocks for unstable attractors with no external explosions leads quickly to sharp results about their shapes and some hints on how Conley's index is related to stability. Then the setting is specialized to the case of flows in ℝⁿ, where unstable attractors are seen to be dynamically complex since they must have external explosions.

Topology and measure of buried points in Julia sets

Clinton P. Curry, John C. Mayer, E. D. Tymchatyn (2013)

Fundamenta Mathematicae

It is well-known that the set of buried points of a Julia set of a rational function (also called the residual Julia set) is topologically “fat” in the sense that it is a dense G δ if it is non-empty. We show that it is, in many cases, a full-measure subset of the Julia set with respect to conformal measure and the measure of maximal entropy. We also address Hausdorff dimension of buried points in the same cases, and discuss connectivity and topological dimension of the set of buried points. Finally,...

Tree algebra of sofic tree languages

Nathalie Aubrun, Marie-Pierre Béal (2014)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We consider the languages of finite trees called tree-shift languages which are factorial extensible tree languages. These languages are sets of factors of subshifts of infinite trees. We give effective syntactic characterizations of two classes of regular tree-shift languages: the finite type tree languages and the tree languages which are almost of finite type. Each class corresponds to a class of subshifts of trees which is invariant by conjugacy. For this goal, we define a tree algebra which...

Two commuting maps without common minimal points

Tomasz Downarowicz (2011)

Colloquium Mathematicae

We construct an example of two commuting homeomorphisms S, T of a compact metric space X such that the union of all minimal sets for S is disjoint from the union of all minimal sets for T. In other words, there are no common minimal points. This answers negatively a question posed in [C-L]. We remark that Furstenberg proved the existence of "doubly recurrent" points (see [F]). Not only are these points recurrent under both S and T, but they recur along the same sequence of powers. Our example shows...

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