The bitransitive continuous maps of the interval are conjugate to maps extremely chaotic a. e.
The Box Dimension of Completely Invariant Subsets for Expanding Piecewise Monotonic Transformations.
The continuous solutions of a generalized Dhombres functional equation
We consider the functional equation where is a given increasing homeomorphism of an open interval and is an unknown continuous function. In a series of papers by P. Kahlig and J. Smítal it was proved that the range of any non-constant solution is an interval whose end-points are fixed under and which contains in its interior no fixed point except for . They also provide a characterization of the class of monotone solutions and prove a necessary and sufficient condition for any solution...
The converse problem for a generalized Dhombres functional equation
We consider the functional equation where is a given homeomorphism of an open interval and is an unknown continuous function. A characterization of the class of continuous solutions is given in a series of papers by Kahlig and Smítal 1998–2002, and in a recent paper by Reich et al. 2004, in the case when is increasing. In the present paper we solve the converse problem, for which continuous maps , where is an interval, there is an increasing homeomorphism of such that . We...
The Covering Principle for Darboux Baire 1 functions
We show that the Covering Principle known for continuous maps of the real line also holds for functions whose graph is a connected subset of the plane. As an application we find an example of an approximately continuous (hence Darboux Baire 1) function f: [0,1] → [0,1] such that any closed subset of [0,1] can be translated so as to become an ω-limit set of f. This solves a problem posed by Bruckner, Ceder and Pearson [Real Anal. Exchange 15 (1989/90)].
The dynamics of complex polynomials and automorphisms of the shift.
The Feigenbaum functional equation and periodic points.
The Feigenbaum functional equation and periodic points. (Short Communication).
The fractal dimension of invariant subsets for piecewise monotonic maps on the interval.
The fundamental theorem of dynamical systems
We propose the title of The Fundamental Theorem of Dynamical Systems for a theorem of Charles Conley concerning the decomposition of spaces on which dynamical systems are defined. First, we briefly set the context and state the theorem. After some definitions and preliminary results, based both on Conley's work and modifications to it, we present a sketch of a proof of the result in the setting of the iteration of continuous functions on compact metric spaces. Finally, we claim that this theorem...
The iterates are not dense in .
The real 3x+1 problem
The stability of Markov operators on Polish spaces
A sufficient condition for the asymptotic stability of Markov operators acting on measures defined on Polish spaces is presented.
The structure of iteration groups of continuous functions. (Summary).
The structure of -limit sets for continuous maps of the interval
We prove that every infinite nowhere dense compact subset of the interval is an -limit set of homoclinic type for a continuous function from to .
Topological conjugacies of piecewise monotone interval maps.
Topological invariance of the Collet–Eckmann property for S-unimodal maps
We prove that if f, g are smooth unimodal maps of the interval with negative Schwarzian derivative, conjugated by a homeomorphism of the interval, and f is Collet-Eckmann, then so is g.
Triangular maps with the chain recurrent points periodic.
Turbulent maps and their ω-limit sets
One-dimensional turbulent maps can be characterized via their ω-limit sets [1]. We give a direct proof of this characterization and get stronger results, which allows us to obtain some other results on ω-limit sets, which previously were difficult to prove.
Two kinds of chaos and relations between them.