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The classification of circle-like continua that admit expansive homeomorphisms

Christopher Mouron (2011)

Fundamenta Mathematicae

A homeomorphism h: X → X of a compactum X is expansive provided that for some fixed c > 0 and every x, y ∈ X (x ≠ y) there exists an integer n, dependent only on x and y, such that d(hⁿ(x),hⁿ(y)) > c. It is shown that if X is a solenoid that admits an expansive homeomorphism, then X is homeomorphic to a regular solenoid. It can then be concluded that a circle-like continuum admits an expansive homeomorphism if and only if it is homeomorphic to a regular solenoid.

The Conley index and countable decompositions of invariant sets

Marian Gidea (1999)

Banach Center Publications

We define a new cohomological index of Conley type associated to any bi-infinite sequence of neighborhoods that satisfies a certain isolation condition. We use this index to study the chaotic dynamics on invariant sets which decompose as countable unions of pairwise disjoint (mod 0) compact pieces.

The Conley index for flows preserving generalized symmetries

Artur Pruszko (1999)

Banach Center Publications

Topological spaces with generalized symmetries are defined and extensions of the Conley index of a compact isolated invariant set of the flow preserving the structures introduced are proposed. One of the two new indexes is constructed with no additional assumption on the examined set in terms of symmetry invariance.

The Conley index theory: A brief introduction

Konstantin Mischaikow (1999)

Banach Center Publications

A brief introduction to the Conley index theory is presented. The emphasis is the fundamental ideas of Conley's approach to dynamical systems and how it avoids some of the difficulties inherent in the study of nonlinear systems.

The entropy conjecture for diffeomorphisms away from tangencies

Gang Liao, Marcelo Viana, Jiagang Yang (2013)

Journal of the European Mathematical Society

We prove that every C 1 diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shub’s entropy conjecture: the entropy is bounded from below by the spectral radius in homology. Moreover, they admit principal symbolic extensions, and the topological entropy and metrical entropy vary continuously with the map. In contrast, generic diffeomorphisms with persistent tangencies are not entropy expansive.

The entropy of algebraic actions of countable torsion-free abelian groups

Richard Miles (2008)

Fundamenta Mathematicae

This paper is concerned with the entropy of an action of a countable torsion-free abelian group G by continuous automorphisms of a compact abelian group X. A formula is obtained that expresses the entropy in terms of the Mahler measure of a greatest common divisor, complementing earlier work by Einsiedler, Lind, Schmidt and Ward. This leads to a uniform method for calculating entropy whenever G is free. In cases where these methods do not apply, a possible entropy formula is conjectured. The entropy...

The geometry of laminations

Robbert Fokkink, Lex Oversteegen (1996)

Fundamenta Mathematicae

A lamination is a continuum which locally is the product of a Cantor set and an arc. We investigate the topological structure and embedding properties of laminations. We prove that a nondegenerate lamination cannot be tree-like and that a planar lamination has at least four complementary domains. Furthermore, a lamination in the plane can be obtained by a lakes of Wada construction.

The Language of Caring: Quantitating Medical Practice Patterns using Symbolic Dynamics

J. Paladino, A. M. Kaynar, P. S. Crooke, J. R. Hotchkiss (2010)

Mathematical Modelling of Natural Phenomena

Real-world medical decisions rarely involve binary Ðsole condition present or absent- patterns of patient pathophysiology. Similarly, provider interventions are rarely unitary in nature: the clinician often undertakes multiple interventions simultaneously. Conventional approaches towards complex physiologic derangements and their associated management focus on the frequencies of joint appearances, treating the individual derangements of physiology...

The nonexistence of universal metric flows

Stefan Geschke (2018)

Commentationes Mathematicae Universitatis Carolinae

We consider dynamical systems of the form ( X , f ) where X is a compact metric space and f : X → X is either a continuous map or a homeomorphism and provide a new proof that there is no universal metric dynamical system of this kind. The same is true for metric minimal dynamical systems and for metric abstract ω -limit sets, answering a question by Will Brian.

The number of binary rotation words

A. Frid, D. Jamet (2014)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We consider binary rotation words generated by partitions of the unit circle to two intervals and give a precise formula for the number of such words of length n. We also give the precise asymptotics for it, which happens to be Θ(n4). The result continues the line initiated by the formula for the number of all Sturmian words obtained by Lipatov [Problemy Kibernet. 39 (1982) 67–84], then independently by Mignosi [Theoret. Comput. Sci. 82 (1991) 71–84], and others.

The omega limit sets of subsets in a metric space

Changming Ding (2005)

Czechoslovak Mathematical Journal

In this paper, we discuss the properties of limit sets of subsets and attractors in a compact metric space. It is shown that the ω -limit set ω ( Y ) of Y is the limit point of the sequence { ( C l Y ) · [ i , ∞ ) } i = 1 ∞ in 2 X and also a quasi-attractor is the limit point of attractors with respect to the Hausdorff metric. It is shown that if a component of an attractor is not an attractor, then it must be a real quasi-attractor.

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