We give a brief introduction to the Bernoulli shift map as a basic chaotic dynamical system. We give several examples where the iterates of a~mapping can be understood using the Bernoulli shift. Namely, the iteration of real interval maps and iteration of quadratic functions in the complex plain.
We define a new cohomological index of Conley type associated to any bi-infinite sequence of neighborhoods that satisfies a certain isolation condition. We use this index to study the chaotic dynamics on invariant sets which decompose as countable unions of pairwise disjoint (mod 0) compact pieces.
We show that for critically non-recurrent rational functions all the definitions of topological pressure proposed in [12] coincide for all t ≥ 0. Then we study in detail the Gibbs states corresponding to the potentials -tlog|f'| and their σ-finite invariant versions. In particular we provide a sufficient condition for their finiteness. We determine the escape rates of critically non-recurrent rational functions. In the presence of parabolic points we also establish a polynomial rate of appropriately...
A continuous map of the interval is chaotic iff there is an increasing sequence of nonnegative integers such that the topological sequence entropy of relative to , , is positive ([FS]). On the other hand, for any increasing sequence of nonnegative integers there is a chaotic map of the interval such that ([H]). We prove that the same results hold for maps of the circle. We also prove some preliminary results concerning topological sequence entropy for maps of general compact metric...
We prove that for continuous interval maps the existence of a non-empty closed invariant subset which is transitive and sensitive to initial conditions is implied by positive topological entropy and implies chaos in the sense of Li-Yorke, and we exhibit examples showing that these three notions are distinct.
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.