Sergiy Kolyada | Institute of mathematics NASU, Kyiv (original) (raw)

Papers by Sergiy Kolyada

Abstract We introduce and study a concept which links the Li–Yorke versions of chaos with the not... more Abstract We introduce and study a concept which links the Li–Yorke versions of chaos with the notion of sensitivity to initial conditions. We say that a dynamical system (X, T) is Li–Yorke sensitive if there exists a positive ε such that every x∈ X is a limit of points y∈ X such that the pair (x, y) is proximal but not ε-asymptotic, ie for infinitely many positive integers i the distance ρ (T i (x), T i (y)) is greater than ε but for any positive δ this distance is less than δ for infinitely many i.

Abstract. The Li–Yorke definition of chaos proved its value for interval maps. In this paper it i... more Abstract. The Li–Yorke definition of chaos proved its value for interval maps. In this paper it is considered in the setting of general topological dynamics. We adopt two opposite points of view. On the one hand sufficient conditions for Li–Yorke chaos in a topological dynamical system are given. We solve a long–standing open question by proving that positive entropy implies Li–Yorke chaos.

Summary. The aim of this paper is to give an axiomatic definition of the topological entropy for ... more Summary. The aim of this paper is to give an axiomatic definition of the topological entropy for continuous interval maps and, in such a way, to shed some more light on the importance of the different properties of the topological entropy in this setting. We give two closely related axiomatic definitions of topological entropy and an axiomatic characterization of the topological chaos.

Abstract If (X, f) is a dynamical system given by a compact metric space X and a continuous map f... more Abstract If (X, f) is a dynamical system given by a compact metric space X and a continuous map f: X→ X then by the functional envelope of (X, f) we mean the dynamical system (S (X), Ff) whose phase space S (X) is the space of all continuous selfmaps of X and the map Ff: S (X)→ S (X) is defined by Ff (ϕ)= f◦ ϕ for any ϕ∈ S (X). The functional envelope of a system always contains a copy of the original system.

Abstract: The aim of this paper is to investigate the connection between transitivity, density of... more Abstract: The aim of this paper is to investigate the connection between transitivity, density of the set of periodic points and topological entropy for low dimensional continuous maps. The paper deals with this problem in the case of the $ n $-star and the circle among the one-dimensional spaces and in some higher dimensional spaces. Particular attention is paid to triangular maps and to extensions of transitive maps to higher dimensions without increasing topological entropy.

Abstract: Topological structure of minimal sets is studied for a dynamical system (E,F)(E, F) (E,F) given... more Abstract: Topological structure of minimal sets is studied for a dynamical system (E,F)(E, F) (E,F) given by a fibre-preserving, in general non-invertible, continuous selfmap $ F $ of a graph bundle $ E .Thesesystemsinclude,asaveryparticularcase,quasiperiodicallyforcedcirclehomeomorphisms.Let. These systems include, as a very particular case, quasiperiodically forced circle homeomorphisms. Let .Thesesystemsinclude,asaveryparticularcase,quasiperiodicallyforcedcirclehomeomorphisms.Let M $ be a minimal set of $ F $ with full projection onto the base space $ B $ of the bundle.

The aim of this paper is to investigate the connection between various definitions of chaos for t... more The aim of this paper is to investigate the connection between various definitions of chaos for topological dynamical systems (i.e., continuous surjective maps of infinite compact metric spaces without isolated points into itself). Particular attention is paid to a very recent definition of spatiotemporal chaos which is based on Li-Yorke pairs and has some common features with sensitivity. We show that all topologically mixing systems, weakly mixing minimal systems, proximal systems and also some classes of recurrent systems are spatiotemporally chaotic.

Scholarpedia, Jan 1, 2009

The concept of topological transitivity goes back to GD Birkhoff [1] who introduced it in 1920 (f... more The concept of topological transitivity goes back to GD Birkhoff [1] who introduced it in 1920 (for flows). This article will concentrate on topological transitivity of dynamical systems given by continuous mappings in metric spaces. Intuitively, a topologically transitive dynamical system has points which eventually move under iteration from one arbitrarily small open set to any other. Consequently, such a dynamical system cannot be decomposed into two disjoint sets with nonempty interiors which do not interact under the transformation.

ABSTRACT This volume contains papers from the special program and international conference on Dyn... more ABSTRACT This volume contains papers from the special program and international conference on Dynamical Numbers which were held at the Max-Planck Institute in Bonn, Germany in 2009. These papers reflect the extraordinary range and depth of the interactions between ergodic theory and dynamical systems and number theory. Topics covered in the book include stationary measures, systems of enumeration, geometrical methods, spectral methods, and algebraic dynamical systems. http://www.ams.org/bookstore?fn=20&arg1=conmseries&ikey=CONM-532

Scholarpedia, Jan 1, 2009

This volume contains a collection of articles from the special program on algebraic and topologic... more This volume contains a collection of articles from the special program on algebraic and topological dynamics and a workshop on dynamical systems held at the Max-Planck Institute (Bonn, Germany). It reflects the extraordinary vitality of dynamical systems in its interaction with a broad range of mathematical subjects. Topics covered in the book include asymptotic geometric analysis, transformation groups, arithmetic dynamics, complex dynamics, symbolic dynamics, statistical properties of dynamical systems, and the theory of entropy ...

Abstract We introduce and study a concept which links the Li–Yorke versions of chaos with the not... more Abstract We introduce and study a concept which links the Li–Yorke versions of chaos with the notion of sensitivity to initial conditions. We say that a dynamical system (X, T) is Li–Yorke sensitive if there exists a positive ε such that every x∈ X is a limit of points y∈ X such that the pair (x, y) is proximal but not ε-asymptotic, ie for infinitely many positive integers i the distance ρ (T i (x), T i (y)) is greater than ε but for any positive δ this distance is less than δ for infinitely many i.

Abstract. The Li–Yorke definition of chaos proved its value for interval maps. In this paper it i... more Abstract. The Li–Yorke definition of chaos proved its value for interval maps. In this paper it is considered in the setting of general topological dynamics. We adopt two opposite points of view. On the one hand sufficient conditions for Li–Yorke chaos in a topological dynamical system are given. We solve a long–standing open question by proving that positive entropy implies Li–Yorke chaos.

Summary. The aim of this paper is to give an axiomatic definition of the topological entropy for ... more Summary. The aim of this paper is to give an axiomatic definition of the topological entropy for continuous interval maps and, in such a way, to shed some more light on the importance of the different properties of the topological entropy in this setting. We give two closely related axiomatic definitions of topological entropy and an axiomatic characterization of the topological chaos.

Abstract If (X, f) is a dynamical system given by a compact metric space X and a continuous map f... more Abstract If (X, f) is a dynamical system given by a compact metric space X and a continuous map f: X→ X then by the functional envelope of (X, f) we mean the dynamical system (S (X), Ff) whose phase space S (X) is the space of all continuous selfmaps of X and the map Ff: S (X)→ S (X) is defined by Ff (ϕ)= f◦ ϕ for any ϕ∈ S (X). The functional envelope of a system always contains a copy of the original system.

Abstract: The aim of this paper is to investigate the connection between transitivity, density of... more Abstract: The aim of this paper is to investigate the connection between transitivity, density of the set of periodic points and topological entropy for low dimensional continuous maps. The paper deals with this problem in the case of the $ n $-star and the circle among the one-dimensional spaces and in some higher dimensional spaces. Particular attention is paid to triangular maps and to extensions of transitive maps to higher dimensions without increasing topological entropy.

Abstract: Topological structure of minimal sets is studied for a dynamical system (E,F)(E, F) (E,F) given... more Abstract: Topological structure of minimal sets is studied for a dynamical system (E,F)(E, F) (E,F) given by a fibre-preserving, in general non-invertible, continuous selfmap $ F $ of a graph bundle $ E .Thesesystemsinclude,asaveryparticularcase,quasiperiodicallyforcedcirclehomeomorphisms.Let. These systems include, as a very particular case, quasiperiodically forced circle homeomorphisms. Let .Thesesystemsinclude,asaveryparticularcase,quasiperiodicallyforcedcirclehomeomorphisms.Let M $ be a minimal set of $ F $ with full projection onto the base space $ B $ of the bundle.

The aim of this paper is to investigate the connection between various definitions of chaos for t... more The aim of this paper is to investigate the connection between various definitions of chaos for topological dynamical systems (i.e., continuous surjective maps of infinite compact metric spaces without isolated points into itself). Particular attention is paid to a very recent definition of spatiotemporal chaos which is based on Li-Yorke pairs and has some common features with sensitivity. We show that all topologically mixing systems, weakly mixing minimal systems, proximal systems and also some classes of recurrent systems are spatiotemporally chaotic.

Scholarpedia, Jan 1, 2009

The concept of topological transitivity goes back to GD Birkhoff [1] who introduced it in 1920 (f... more The concept of topological transitivity goes back to GD Birkhoff [1] who introduced it in 1920 (for flows). This article will concentrate on topological transitivity of dynamical systems given by continuous mappings in metric spaces. Intuitively, a topologically transitive dynamical system has points which eventually move under iteration from one arbitrarily small open set to any other. Consequently, such a dynamical system cannot be decomposed into two disjoint sets with nonempty interiors which do not interact under the transformation.

ABSTRACT This volume contains papers from the special program and international conference on Dyn... more ABSTRACT This volume contains papers from the special program and international conference on Dynamical Numbers which were held at the Max-Planck Institute in Bonn, Germany in 2009. These papers reflect the extraordinary range and depth of the interactions between ergodic theory and dynamical systems and number theory. Topics covered in the book include stationary measures, systems of enumeration, geometrical methods, spectral methods, and algebraic dynamical systems. http://www.ams.org/bookstore?fn=20&arg1=conmseries&ikey=CONM-532

Scholarpedia, Jan 1, 2009

This volume contains a collection of articles from the special program on algebraic and topologic... more This volume contains a collection of articles from the special program on algebraic and topological dynamics and a workshop on dynamical systems held at the Max-Planck Institute (Bonn, Germany). It reflects the extraordinary vitality of dynamical systems in its interaction with a broad range of mathematical subjects. Topics covered in the book include asymptotic geometric analysis, transformation groups, arithmetic dynamics, complex dynamics, symbolic dynamics, statistical properties of dynamical systems, and the theory of entropy ...