Weakly mixing, topologically weakly mixing, and sensitivity for non-autonomous discrete systems

This paper is concerned with relationships of weakly mixing, topologically weakly mixing, and sensitivity for non-autonomous discrete systems. It is shown that weakly mixing implies topologically weakly mixing and sensitivity for measurable systems with a fully supported measure; and topological weakly mixing implies sensitivity for general dynamical systems. However, the inverse conclusions are not true and some counterexamples are given. The related existing results for autonomous discrete systems are generalized to non-autonomous discrete systems and their conditions are weaken.Comment: 7 page

Exploring F-Sensitivity for Non-Autonomous Systems

We study some stronger forms of sensitivity, namely, F-sensitivity and weakly F-sensitivity for non-autonomous discrete dynamical systems. We obtain a condition under which these two forms of sensitivity are equivalent. We also justify the difference between F-sensitivity and some other stronger forms of sensitivity through examples. We explore the relation between the F-sensitivity of the non-autonomous system (X, f1;infinity) and autonomous system (X, f), where fn is a sequence of continuous functions converging uniformly to f. We also study the F-sensitivity of a non-autonomous system (X, f1;infinity), generated by a finite family of maps F = f1, f2, : : : , fk and give an example showing that such non-autonomous systems can be F-sensitive, even when none of the maps in the family F is F-sensitive

Dynamics of weakly mixing non-autonomous systems

For a commutative non-autonomous dynamical system we show that topological transitivity of the non-autonomous system induced on probability measures (hyperspaces) is equivalent to the weak mixing of the induced systems. Several counter examples are given for the results which are true in autonomous but need not be true in non-autonomous systems. Wherever possible sufficient conditions are obtained for the results to hold true. For a commutative periodic non-autonomous system on intervals, it is proved that weakly mixing implies Devaney chaos. Given a periodic non-autonomous system, it is shown that sensitivity is equivalent to some stronger forms of sensitivity on a closed unit interval.Comment: 17 page

Lyapunov exponents, sensitivity, and stability for non-autonomous discrete systems

This paper is concerned with relationships of Lyapunov exponents with sensitivity and stability for non-autonomous discrete systems. Some new concepts are introduced for non-autonomous discrete systems, including Lyapunov exponents, strong sensitivity at a point and in a set, Lyapunov stability, and exponential asymptotical stability. It is shown that the positive Lyapunov exponent at a point implies strong sensitivity for a class of non-autonomous discrete systems. Furthermore, the uniformly positive Lyapunov exponents in a totally invariant set imply strong sensitivity in this set under certain conditions. It is also shown that the negative Lyapunov exponent at a point implies exponential asymptotical stability for a class of non-autonomous discrete systems. The related existing results for autonomous discrete systems are generalized to non-autonomous discrete systems and their conditions are weaken. One example is provided for illustration.Comment: 11 page

Furstenberg families and transitivity in non-autonomous systems

We obtain necessary and sufficient conditions for a non-autonomous system to be $\mathcal{F}$-transitive and $\mathcal{F}$-mixing, where $\mathcal{F}$ is a Furstenberg family. We also obtain some characterizations for topologically ergodic non-autonomous systems. We provide examples/counter examples related to our results.Comment: arXiv admin note: text overlap with arXiv:1806.0069

Dynamics of Nonautonomous Discrete Dynamical Systems

In this paper we study the dynamics of a general non-autonomous dynamical system generated by a family of continuous self maps on a compact space $X$. We derive necessary and sufficient conditions for the system to exhibit complex dynamical behavior. In the process we discuss properties like transitivity, weakly mixing, topologically mixing, minimality, sensitivity, topological entropy and Li-Yorke chaoticity for the non-autonomous system. We also give examples to prove that the dynamical behavior of the non-autonomous system in general cannot be characterized in terms of the dynamical behavior of its generating functions

Devaney chaos in non-autonomous discrete systems

This paper is concerned with Devaney chaos in non-autonomous discrete systems. It is shown that in its definition, the two former conditions, i.e., transitivity and density of periodic points, in a set imply the last one, i.e., sensitivity, in the case that the set is unbounded, while a similar result holds under two additional conditions in the other case that the set is bounded. Furthermore, some chaotic behavior is studied for a class of non-autonomous systems, each of which is governed by a convergent sequence of continuous maps.Comment: 13 page

On Dynamics Generated by a Uniformly Convergent Sequence of Maps

In this paper, we study the dynamics of a non-autonomous dynamical system $(X,\mathbb{F})$ generated by a sequence $(f_n)$ of continuous self maps converging uniformly to $f$. We relate the dynamics of the non-autonomous system $(X,\mathbb{F})$ with the dynamics of $(X,f)$. We prove that if the family $\mathbb{F}$ commutes with $f$ and $(f_n)$ converges to $f$ at a "sufficiently fast rate", many of the dynamical properties for the systems $(X,\mathbb{F})$ and $(X,f)$ coincide. In the procees we establish equivalence of properties like equicontinuity, minimality and denseness of proximal pairs (cells) for the two systems. In addition, if $\mathbb{F}$ is feeble open, we establish equivalence of properties like transitivity, weak mixing and various forms of sensitivities. We prove that feeble openness of $\mathbb{F}$ is sufficient to establish equivalence of topological mixing for the two systems. We prove that if $\mathbb{F}$ is feeble open, dynamics of the non-autonomous system on a compact interval exhibits any form of mixing if and only if $(X,f)$ exhibits identical form of mixing. We also investigate dense periodicity for the two systems. We give examples to investigate sufficiency/necessity of the conditions imposed. In the process we derive weaker conditions under which the established dynamical relation (between the two systems $(X,\mathbb{F})$ and $(X,f)$) is preserved

Dynamics Of Finitely Generated Non-Autonomous Systems

In this paper, we discuss dynamical behavior of a non-autonomous system generated by a finite family $\mathbb{F}$. In the process, we relate the dynamical behavior of the non-autonomous system generated by the family $\mathbb{F}=\{f_1,f_2,\ldots,f_k\}$ with the dynamical behavior of the system $(X,f_k\circ f_{k-1}\circ\ldots\circ f_1)$. We discuss properties like minimality, equicontinuity, proximality and various forms of sensitivities for the two systems. We derive conditions under which the dynamical behavior of $(X,f_k\circ f_{k-1}\circ\ldots\circ f_1)$ is carried forward to $(X,\mathbb{F})$ (and vice-versa). We also give examples to illustrate the necessity of the conditions imposed

Alterations And Rearrangements Of A Non-Autonomous Dynamical System

In this paper, we discuss the dynamics of alterations and rearrangements of a non-autonomous dynamical system generated by the family $\mathbb{F}$. We prove that while insertion/deletion of a map in the family $\mathbb{F}$ can disturb the dynamics of a system, the dynamics of the system does not change if the map inserted/deleted is feeble open. In the process, we prove that if the inserted/deleted map is feeble open, the altered system exhibits any form of mixing/sensitivity if and only if the original system exhibits the same. We extend our investigations to properties like equicontinuity, minimality and proximality for the two systems. We prove that any finite rearrangement of a non-autonomous dynamical system preserves the dynamics of original system if the family $\mathbb{F}$ is feeble open. We also give examples to show that the dynamical behavior of a system need be not be preserved under infinite rearrangement