We prove that if A is a basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, then the periodic points in the boundary of A are dense in this boundary. To prove this in the non-simply connected or parabolic situations we prove a more abstract, geometric coding trees version.

This paper is concerned with distributional chaos of time-varying discrete systems in metric spaces. Some basic concepts are introduced for general time-varying systems, including sequentially distributive chaos, weak mixing, and mixing. We give an example of sequentially distributive chaos of finite-dimensional linear time-varying dynamical systems, which is not distributively chaotic of type i (DCi for short, i = 1, 2). We also prove that two uniformly topological equiconjugate time-varying systems...

We show that two permutable transcendental entire functions may have different dynamical properties, which is very different from the rational functions case.

The paper is concerned with the dynamics of an entire transcendental function whose inverse has only finitely many singularities. It is rpoven that there are no escaping orbits on the Fatou set. Under some extra assumptions the set of escaping orbits has zero Lebesgue measure. If a function depends analytically on parameters then a periodic point as a function of parameters has only algebraic singularities. This yields the Structural Stability Theorem.

We prove complex bounds for infinitely renormalizable real quadratic maps with essentially bounded combinatorics. This is the last missing ingredient in the problem of complex bounds for all infinitely renormalizable real quadratics. One of the corollaries is that the Julia set of any real quadratic map $z\mapsto z^2+c$, $c\in [-2,1/4]$, is locally connected.