arXiv subjects

Sergii Kolyada

Publications and source records attributed to Sergii Kolyada.

4 recordsLinked to original sources

arXiv2016

Auslander-Yorke dichotomy theorem, multi-sensitivity and Lyapunov numbers

In this paper we study several stronger forms of sensitivity for continuous surjective selfmaps on compact metric spaces and relations between them. The main result of the paper states that a minimal system is either multi-sensitive or an almost one-to-one extension of its maximal equicontinuous factor, which is an analog of the Auslander-Yorke dichotomy theorem. For minimal dynamical systems, we also show that all notions of thick sensitivity, multi-sensitivity and thickly syndetical sensitivity are equivalent, and all of them are much stronger than sensitivity.

math.DS↗

arXiv2016

Dynamical compactness and sensitivity

To link the Auslander point dynamics property with topological transitivity, in this paper we introduce dynamically compact systems as a new concept of a chaotic dynamical system $(X,T)$ given by a compact metric space $X$ and a continuous surjective self-map $T:X \to X$. Observe that each weakly mixing system is transitive compact, and we show that any transitive compact M-system is weakly mixing. Then we discuss the relationships among it and other several stronger forms of sensitivity. We prove that any transitive compact system is Li-Yorke sensitive and furthermore multi-sensitive if it is not proximal, and that any multi-sensitive system has positive topological sequence entropy. Moreover, we show that multi-sensitivity is equivalent to both thick sensitivity and thickly syndetic sensitivity for M-systems. We also give a quantitative analysis for multi-sensitivity of a dynamical system.

math.DS↗

arXiv2016

Analogues of Auslander-Yorke theorems for multi-sensitivity

In this paper we study multi-sensitivity and thick sensitivity for continuous surjective selfmaps on compact metric spaces. We show that multi-sensitivity implies thick sensitivity, and the converse holds true for transitive systems. Our main result is an analog of the Auslander-Yorke dichotomy theorem: a minimal system is either multi-sensitive or an almost one-to-one extension of its maximal equicontinuous factor. Furthermore, we refine it by introducing the concept of syndetically equicontinuous points: a transitive system is either thickly sensitive or contains syndetically equicontinuous points.

math.DS↗

arXiv2013

Minimal sets of fibre-preserving maps in graph bundles

Topological structure of minimal sets is studied for a dynamical system $(E,F)$ given by a fibre-preserving, in general non-invertible, continuous selfmap $F$ of a graph bundle $E$. These systems include, as a very particular case, quasiperiodically forced circle homeomorphisms. Let $M$ be a minimal set of $F$ with full projection onto the base space $B$ of the bundle. We show that $M$ is nowhere dense or has nonempty interior depending on whether the set of so called endpoints of $M$ is dense in $M$ or is empty. If $M$ is nowhere dense, we prove that either a typical fibre of $M$ is a Cantor set, or there is a positive integer $N$ such that a typical fibre of $M$ has cardinality $N$. If $M$ has nonempty interior we prove that there is a positive integer $m$ such that a typical fibre of $M$, in fact even each fibre of $M$ over a \emph{dense open} set $\mathcal O \subseteq B$, is a disjoint union of $m$ circles. Moreover, we show that each of the fibres of $M$ over $B\setminus \mathcal O$ is a union of circles properly containing a disjoint union of $m$ circles. Surprisingly, some of the circles in such "non-typical" fibres of $M$ may intersect. We also give sufficient conditions for $M$ to be a sub-bundle of $E$.

math.DS↗