Marta Štefánková

Specification property and distributional chaos almost everywhere

Our main result shows that a continuous map f acting on a compact metric space (X, p) with a weaker form of specification property and with a pair of distal points is distributionally chaotic in a very strong sense. Strictly speaking, there is a distributionally scrambled set S dense in X which is the union of disjoint sets homeomorphic to Cantor sets so that, for any two distinct points u, v S, the upper distribution function is identically 1 and the lower distribution function is zero at some £ > 0. As a consequence, we describe a class of maps with a scrambled set of full Lebesgue measure in the case when X is the k-dimensional cube I k . If X = I, then we can even construct scrambled sets whose complements have zero Hausdorff dimension.

Functional equation of Dhombres type in the real case

We consider continuous solutions f : R+ ! R+ = (0,1) of the functional equation f(xf(x)) = o(f(x)) where o is a given continuous map R+ ! R+. If o is an increasing homeomorphism the solutions are completely described, if not there are only partial results. In this paper we bring some necessary conditions upon a possible range Rf . In particular, if ojRf has no periodic points except for xed points then there are at most two xed points in Rf , and all possible types of Rf and all possible types of behavior of f can be described. The paper contains techniques which essentially simplify the description of the class of all solutions.

Strong and weak distributional chaos

The original notion of distributional chaos introduced in 1994 by Schweizer and Smítal for continuous maps of the interval was later generalized to arbitrary compact metric space, and three types, DC1–DC3, are now considered. However, most of the results concern the case when the scrambled set consists of two points, . In this paper, we consider stronger versions of distributional chaos, , where uncountable scrambled set is required. We show, among others, that these types and are mutually non-equivalent, even in the class of triangular maps of the square.

On generic and dense chaos for maps induced on hyperspaces

A continuous map f on a compact metric space X induces in a natural way the map on the hyperspace of all closed non-empty subsets of X. We study the question of transmission of chaos between f and . We deal with generic, generic -, dense and dense -chaos for interval maps. We prove that all four types of chaos transmit from f to , while the converse transmission from to f is true for generic, generic - and dense -chaos. Moreover, the transmission of dense - and generic -chaos from to f is true for maps on general compact metric spaces.

Generalized Dhombres Functional Equation

We consider the equation f(xf(x)) = φ(f(x)), x > 0, where φ is given, and f is an unknown continuous function (0, ∞) → (0, ∞). This equation was for the first time studied in 1975 by Dhombres (with φ(y) = y2), later it was considered for other particular choices of φ, and since 2001 for arbitrary continuous function φ. The main problem, a classification of possible solutions and a description of the structure of periodic points contained in the range of the solutions (which appeared to be important way of the classification of solutions), was basically solved. This process involved not only methods from one-dimensional dynamics but also some new methods which could be useful in other problems. In this paper we provide a brief survey.

On -limit sets of non-autonomous dynamical systems with a uniform limit of type

This paper is devoted to the study of properties of -limit sets of non-autonomous dynamical systems on compact metric spaces given by sequences of maps which uniformly converge to a continuous map f. We show that, for systems defined on compact metric spaces, if an -limit set of the non-autonomous system is a subset of the set P(f) of periodic points of f then is necessarily the union of finitely many disjoint connected sets which are cyclically mapped to one another. Using this result, we answer a question posed by Cánovas in [3] [On -limit sets of non-autonomous systems. J. Difference Equ. Appl. 12 (2006), pp. 95–100] by proving that, if an interval map f has only finite -limit sets, then any -limit set of the non-autonomous system is a subset of the set of periodic points of f. We also show that a similar result applies to systems on trees but not on graphs with loops.