J. Smítal

Measures of chaos and a spectral decomposition of dynamical systems on the interval

Let f: [0, 1] -+ [0, 1] be continuous. For x, y E [0, 1], the upper and lower (distance) distribution functions, Fx*y and Fxy, are defined for any t > 0 as the lim sup and lim inf as n -+ oc of the average number of times that the distance Ifi (x) fi(y)I between the trajectories of x and y is less than t during the first n iterations. The spectrum of f is the system ?(f) of lower distribution functions which is characterized by the following properties: (1) The elements of ?(f) are mutually incomparable; (2) for any F E ?(f), there is a perfect set PF #8 0 such that FUV = F and FUV 1 for any distinct u, v E PF; (3) if S is a scrambled set for f, then there are F, G in ?(f) and a decomposition S = SF U SG (SG may be empty) such that FUV > F if u, v E SF and FUV > G if u, v E SG . Our principal results are: (1) If f has positive topological entropy, then ?(f) is nonempty and finite, and any F E ?(f) is zero on an interval [0, e], where e > 0 (and hence any PF is a scrambled set in the sense of Li and Yorke). (2) If f has zero topological entropy, then ?(f) = {F} where F 1. It follows that the spectrum of f provides a measure of the degree of chaos of f . In addition, a useful numerical measure is the largest of the numbers fo(1 F(t))dt, where F E ?(f).

Chaotic functions with zero topological entropy

On caracterise la classe M⊂C°(I,I) des applications chaotiques en ce sens. On montre que M contient certaines appliquations qui ont a la fois une entropie topologique nulle et des attracteurs infinis. On montre que le complement de M consiste en applications qui ont seulement des trajectoires approchables par des cycles

CHARACTERIZATIONS OF WEAKLY CHAOTIC MAPS OF THE INTERVAL

(Communicated by R. Daniel Mauldin)Abstract. We prove, among others, the following relations between notionsof chaos for continuous maps of the interval: (i) A map / is not chaotic inthe sense of Li and Yorke iff / restricted to the set of its w-limit points isstable in the sense of Ljapunov. (ii) The topological entropy of / is zero iff/ restricted to the set of chain recurrent points is not chaotic in the sense ofLi and Yorke, and this is iff every trajectory is approximable by trajectories ofperiodic intervals.

The space of -limit sets of a continuous map of the interval

We first give a geometric characterization of ω-limit sets. We then use this characterization to prove that the family of ω-limit sets of a continuous interval map is closed with respect to the Hausdorff metric. Finally, we apply this latter result to other dynamical systems.

STRANGE TRIANGULAR MAPS OF THE SQUARE

We show that continuous triangular maps of the square I 1 , F : ( x, y ) → ( f ( x ), g ( x, y )), exhibit phenomena impossible in the one-dimensional case. In particular: (1) A triangular map F with zero topological entropy can have a minimal set containing an interval { a } × I , and can have recurrent points that are not uniformly recurrent; this solves two problems by S.F. Kolyada. (2) In the class of mappings satisfying Per( F ) = Fix( F ), there are non-chaotic maps with positive sequence topological entropy and chaotic maps with zero sequence topological entropy.

A chaotic function with a scrambled set of positive Lebesgue measure

There is a continuous map of the unit interval / which is chaotic in the sense of Li and Yorke and which has a scrambled set of Lebesgue measure arbitrarily close to 1. If / is a continuous self-mapping of 7 = (0,1) then by fn we always denote the nth iterate of /. We say that {fn(x)}n<Lx, for any x G 7, is a sequence generated by x; it is said to be asymptotically periodic provided its cluster set is finite. Further, / is chaotic if there is an uncountable set S E I (a scrambled set) of points which generate sequences not asymptotically periodic and such that, for every x,y E S,

ON THE STRUCTURE OF THE ω-LIMIT SETS FOR CONTINUOUS MAPS OF THE INTERVAL

We introduce the notion of the center of a point for discrete dynamical systems and we study its properties for continuous interval maps. It is known that the Birkhoff center of any such map has depth at most 2. Contrary to this, we show that if a map has positive topological entropy then, for any countable ordinal α, there is a point xα∈I such that its center has depth at least α. This improves a result by [Sharkovskii, 1966].

Iteration Theory: Dynamical Systems and Functional Equations

The aim of this paper is to give an account of some problems considered in the past years in the setting of Discrete Dynamical Systems and Iterative Functional Equations, some new research directions and also state some open problems.

On a generalized Dhombres functional equation

Summary. We consider the functional equation

Why it is important to understand dynamics of triangular maps

f(xf(x))=\varphi (f(x))