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Ruchi Das

Publications and source records attributed to Ruchi Das.

12 recordsLinked to original sources
arXiv2026

Proximal Relations and Maximal Equicontinuous Factors for Non-autonomous Dynamical Systems

This paper studies proximal-type relations and maximal equicontinuous factors for non-autonomous dynamical systems on compact metric spaces. Two classes of systems are considered: systems generated by uniformly convergent sequences of maps and periodic systems. For uniformly convergent generators, collective convergence and uniform rigidity imply that the regional proximal relation for the non-autonomous system is equal to the regional proximal relation for the limit autonomous system and if the limit system is equicontinuous, then the proximality relation for the non-autonomous system is equal to the proximal relation for the autonomous limit system. In the same setting, the kernel of the maximal equicontinuous factor is identified as the smallest closed equivalence relation containing the autonomous equicontinuous structure relation of the corresponding autonomous system and preserved by all generators. For periodic systems, regional proximality and Banach proximality are determined by the period map, yielding a criterion in terms of mean equicontinuity and the relation of sensitivity in the mean. Examples show that the hypotheses used in these results are necessary.

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arXiv2025

Auslander-Yorke Dichotomy and Its Generalizations for Non-Autonomous Dynamical Systems

We investigate the dynamics of periodic non-autonomous discrete dynamical systems on uniform spaces and topological spaces, focusing on the extension of the classical Auslander-Yorke dichotomy to these settings. We prove various dichotomy theorems in the uniform space framework, showing that a minimal periodic non-autonomous system is either sensitive or equicontinuous, and prove some more refined versions involving syndetic equicontinuity and thick sensitivity and eventual sensitivity versus equicontinuity on compact uniform spaces. We further introduce topological analogues like topological equicontinuity, Hausdorff sensitivity, and their syndetic and multi-sensitive variants and prove corresponding Auslander-Yorke-type dichotomies on T3 spaces.

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arXiv2023

Dynamics of multi-sensitive non-autonomous systems with respect to a vector

We introduce the concept of multi-sensitivity with respect to a vector for a non-autonomous discrete system. We prove that for a periodic non-autonomous system on the closed unit interval, sensitivity is equivalent to strong multi-sensitivity and justify that the result need not be true if the system is not periodic. In addition, we study strong multi-sensitivity and N-sensitivity on non-autonomous systems induced by probability measure spaces. Moreover, we first prove that if fn converges to f uniformly, then strong multi-sensitivity (respectively, N-sensitivity) of the non-autonomous system does not coincide with that of (X, f). Then we give a sufficient condition such that non-autonomous system is strongly multi-sensitive (respectively, N-sensitive) if and only if f is so. Finally, we prove that if a non-autonomous system converges uniformly, then multi-transitivity and dense periodicity imply N-sensitivity.

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arXiv2019

Continuum-wise expansive Non-Autonomous Discrete Systems

In this paper, we study continuum-wise expansive non-autonomous discrete dynamical systems. We discuss various properties of such non-autonomous systems. We further obtain results for cw-expansive non-autonomous systems with shadowing property and obtain an important equivalence. \keywords{Non-autonomous dynamical systems, continuum-wise expansive, shadowing property

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arXiv2019

Generalizations of expansiveness in Non-Autonomous Discrete Systems

In this paper, we define and study generalizations of expansiveness, namely n- expansiveness, $\aleph_0$-expansiveness, continuum-wise expansive and meagre expansiveness for non-autonomous discrete dynamical systems. We discuss various properties of such non-autonomous systems and give necessary examples. We prove results related to non-existence of $\aleph_0$-expansive and meagre expansive non-autonomous system on certain spaces. We also study relation between $\aleph_0$-expansive and meagre-expansive non-autonomous systems.

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arXiv2018

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.

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arXiv2018

Specification properties for non-autonomous discrete systems

In this paper notions of strong specification property and quasi-weak specification property for non-autonomous discrete systems are introduced and studied. It is shown that these properties are dynamical properties and are preserved under finite product. It is proved that a k-periodic non-autonomous system on intervals having weak specification is Devaney chaotic whereas if the system has strong specification then the result is true in general. Specification properties of induced systems on hyperspaces and probability measures spaces are also studied. Examples/counter examples are provided wherever necessary to support results obtained.

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arXiv2018

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.

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arXiv2018

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.

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arXiv2018

Induced Dynamics in Hyperspaces of Non-Autonomous Discrete Systems

In this paper, the interrelations of some dynamical properties of the non-autonomous dynamical system (X, f1;infinity) and its induced non-autonomous dynamical system (K(X), f1;infinity) are studied, where K(X) is the hyperspace of all non-empty compact subsets of X, endowed with Vietoris topology. Various stronger forms of sensitivity and transitivity are considered. Some examples of non-autonomous systems are provided to support the results. A relation between shadowing property of the non- autonomous system (X, f1;infinity) and its induced system (K(X), f1;infinity) is studied.

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arXiv2016

Exploring stronger forms of transitivity on G-spaces

In this paper we introduce and study some stronger forms of transitivity like total transitivity, weakly mixing for maps on G-spaces. We obtain their relationship with the earlier defined notion of strongly mixing for maps on G-spaces. We also study G-minimal maps on G-spaces in detail.

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