arXiv subjects

Tarun Das

Publications and source records attributed to Tarun Das.

13 recordsLinked to original sources

arXiv2026

Topological Stability in Paired Dynamical Systems

We study the classical topological dynamical notions of shadowing and topological stability from a viewpoint of paired dynamical system $(f,g)$, where $f$ and $g$ are uniform equivalences on a metric space $X$. We observe that if $g$ is equicontinuous and commutes with $f$, then the study of shadowing for $(f,g)$ is reduced to the study of the classical shadowing for $g^{-1}f$. The fact that these assumptions are sufficient is justified through examples. Finally, we prove that if $f$ is an expansive homeomorphism on a relatively compact metric space, then any pair $(f,g)$ with shadowing is topologically stable.

math.DS↗

arXiv2022

Bi-Asymptotic $c$-Expansivity

In this paper, we define bi-asymptotically $c$-expansive maps on metric spaces and study its relationship with other variants of expansivity such as bi-asymptotically expansive maps and $N$-expansive maps. We also provide an example to establish that expansive homeomorphisms need not be bi-asymptotically expansive. Finally we prove a spectral decomposition theorem for bi-asymptotically $c$-expansive continuous surjective maps with the shadowing property on compact metric spaces.

math.DS↗

arXiv2021

Topologically stable and $β$-persistent points of group actions

In this paper, we introduce topologically stable points, $β$-persistent points, $β$-persistent property, $β$-persistent measures and almost $β$-persistent measures for first countable Hausdorff group actions of compact metric spaces. We prove that the set of all $β$-persistent points is measurable and it is closed if the action is equicontinuous. We also prove that the set of all $β$-persistent measures is a convex set and every almost $β$-persistent measure is a $β$-persistent measure. Finally, we prove that every equicontinuous pointwise topologically stable first countable Hausdorff group action of a compact metric space is $β$-persistent. In particular, every equicontinuous pointwise topologically stable flow is $β$-persistent.

math.DS↗

arXiv2021

Mean Ergodic Shadowing

We introduce and study a new variant of shadowing namely mean ergodic shadowing. We establish relationship of this variant with several other variants of shadowing. We show that a minimal system with shadowing cannot have mean ergodic shadowing. We give a necessary and sufficient condition for an orbital limit function to have mean ergodic shadowing property.

math.DS↗

arXiv2019

Stability Theorems in Pointwise Dynamics

We introduce minimally expansive and GH-stable points for homeomorphisms on metric spaces and $μ$-uniformly expansive, $μ$-shadowable and strong $μ$-topologically stable points for Borel measures (with respect to a homeomorphism on a metric space). We prove that: (i) minimally expansive shadowable point of a homeomorphism on a compact metric space is topologically stable and GH-stable. (ii) $μ$-uniformly expansive $μ$-shadowable point for a Borel measure $μ$ (with respect to a homeomorphism on a compact metric space) is strong $μ$-topologically stable.

math.DS↗

arXiv2019

Pointwise dynamics under Orbital Convergence

We obtain sufficient conditions under which the limit of a sequence of functions exhibits a particular dynamical behaviour at a point like expansivity, shadowing, mixing, sensitivity and transitivity. We provide examples to show that the set of all expansive, positively expansive and sensitive points are neither open nor closed in general. We also observe that the set of all transitive and mixing points are closed but not open in general. We give examples to show that properties like expansivity, sensitivity, shadowing, transitivity and mixing at a point need not be preserved under uniform convergence and properties like topological stability and $α$-persistence at a point need not be preserved under pointwise convergence.

math.DS↗

arXiv2019

Specification for Group Actions on Uniform Spaces

We extend specification and periodic specification to finitely generated group actions on uniform spaces using a concept of specification point. We prove that certain group actions having two distinct specification points have positive entropy. We further prove that if a group containing an infinite order element acts on an infinite Hausdorff uniform space and the action possesses periodic specification, then it is Devaney chaotic.

math.DS↗

arXiv2019

Topologically Stable Equicontinuous Non-Autonomous Systems

We find sufficient conditions for commutative non-autonomous systems on certain metric spaces to be topologically stable. In particular, we prove that (i) Every mean equicontinuous, mean expansive system with strong average shadowing property is topologically stable. (ii) Every equicontinuous, recurrently expansive system with almost shadowing property is topologically stable. (iii) Every equicontinuous, expansive system with shadowing property is topologically stable.

math.DS↗

arXiv2018

GH-stability and Spectral Decomposition for Group Actions

We study expansivity and the shadowing property for finitely generated group actions on metric spaces. We consider the projecting and lifting problems for actions having these properties. We prove that every expansive action with the shadowing property is strongly $GH$-stable (Gromov-Hausdorff stable). Finally, we introduce sequential shadowing property for finitely generated group actions on metric spaces and show that such shadowing is strong enough to imply the spectral decomposition property.

math.DS↗

arXiv2018

Measure Expansivity and Specification for Pointwise Dynamics

We introduce pointwise measure expansivity for bi-measurable maps. We show through examples that this notion is weaker than measure expansivity. In spite of this fact, we show that many results for measure expansive systems hold true for pointwise systems as well. Then, we study the concept of mixing, specification and chaos at a point in the phase space of a continuous map. We show that mixing at a shadowable point is not sufficient for it to be a specification point, but mixing of the map force a shadowable point to be a specification point. We prove that periodic specification points are Devaney chaotic point. Finally, we show that existence of two distinct specification points is sufficient for a map to have positive Bowen entropy.

math.DS↗

arXiv2018

Various Non-autonomous Notions for Borel Measures

We introduce and investigate the notions of expansiveness, topological stability and persistence for Borel measures with respect to time varying bi-measurable maps on metric spaces. We prove that expansive persistent measures are topologically stable in the class of all time varying homeomorphisms.

math.DS↗

arXiv2018

Stability Theorems for Group Actions on Uniform Spaces

We extend the notions of topological stability, shadowing and persistence from homeomorphisms to finitely generated group actions on uniform spaces and prove that an expansive action with either shadowing or persistence is topologically stable. Using the concept of null set of a Borel measure $μ$, we introduce the notions of $μ$-expansivity, $μ$-topological stability, $μ$-shadowing and $μ$-persistence for finitely generated group actions on uniform spaces and show that a $μ$-expansive action with either $μ$-shadowing or $μ$-persistence is $μ$-topologically stable.

math.DS↗

arXiv2011

Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces

We introduce topological definitions of expansivity, shadowing, and chain recurrence for homeomorphisms. They generalize the usual definitions for metric spaces. We prove various theorems about topologically Anosov homeomorphisms (maps that are expansive and have the shadowing property) on noncompact and non-metrizable spaces that generalize theorems for such homeomorphisms on compact metric spaces. The main result is a generalization of Smale's spectral decomposition theorem to topologically Anosov homeomorphisms on first countable locally compact paracompact Hausdorff spaces.

math.DS↗