S. F. Kolyada

Li–Yorke sensitivity

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 e such that every xX is a limit of points yX such that the pair (x,y) is proximal but not e-asymptotic, i.e. for infinitely many positive integers i the distance ρ(Ti(x),Ti(y)) is greater than e but for any positive δ this distance is less than δ for infinitely many i.

On dynamics of triangular maps of the square

The paper is devoted to the triangular maps of the square into itself. The results presented were recently obtained by the author and are briefly stated (in Russian) in a difficult paper as well as those (jointly published with A. N. Sharkovsky) published in ECIT-89 (abstract). All these results are systematized and extended by the new ones. The more detailed proofs of all statements are given. It is shown, for example, that triangular maps exist such that their minimal attraction centres do not coincide with the centres, as well as such ones exist that the Milnor attractor is not contained in the closure of the set of periodic points.

Entropy and periodic points for transitive maps

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.

Functional envelope of a dynamical system

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() = f for any S(X). The functional envelope of a system always contains a copy of the original system.Our motivation for the study of dynamics in functional envelopes comes from semigroup theory, from the theory of functional difference equations and from dynamical systems theory. The paper mainly deals with the connection between the properties of a system and the properties of its functional envelope. Special attention is paid to orbit closures, ω-limit sets, (non)existence of dense orbits and topological entropy.

ON MINIMALITY OF NONAUTONOMOUS DYNAMICAL SYSTEMS

The minimality of a nonautonomous dynamical system given by a compact Hausdorff space X and a sequence of continuous self-maps of X is studied. A sufficient condition for the nonminimality of such a system is formulated. Special attention is given to the particular case where X is a real compact interval I. A sequence of continuous self-maps of I forming a minimal nonautonomous system may converge uniformly. For example, the limit may be any topologically transitive map. However, if all maps in the sequence are surjective, then the limit is necessarily monotone. An example where the limit is the identity is given. As an application, in a simple way we construct a triangular map in the square I2 with the property that every point except those in the leftmost fiber has an orbit whose ω-limit set coincides with the leftmost fiber.

Proper minimal sets on compact connected 2-manifolds are nowhere dense

Let M 2 be a compact connected two-dimensional manifold, with or without boundary, and let f : M 2 ! M 2 be a continuous map. We prove that if M M 2 is a minimal set of the dynamical system (M 2 , f ) then either M = M 2 or M is a nowhere dense subset of M 2 . Moreover, we add a shorter proof of the recent result of Blokh, Oversteegen and Tymchatyn, that in the former case M 2 is a torus or a Klein bottle.

Minimal sets of fibre-preserving maps in graph bundles

Topological structure of minimal sets is studied for a dynamical system

Dynamics of One-Dimensional Maps

(E,F)