arXiv subjects

Sushmita Yadav

Publications and source records attributed to Sushmita Yadav.

3 recordsLinked to original sources

arXiv2026

Proximal Relations in Asymptotically Commutative Non-Autonomous Dynamical Systems

In this paper, we investigate various forms of proximal relations for non-autonomous dynamical systems. In particular, we introduce the notion of pointwise asymptotic commutativity and asymptotic commutativity, and show that these are strictly weaker than classical commutativity. We establish conditions under which different proximal relations, including proximality, syndetic proximality, and regional proximality, exhibit invariance and structural properties. In particular, we relate proximality with equicontinuity and weak mixing, and characterize conditions where proximal relations become trivial or maximal. Further, for systems generated by uniformly and strongly convergent sequences, we derive connections between proximal pairs of the system, its product systems, and the corresponding limiting system. Several examples are provided to demonstrate the necessity of the assumptions and the sharpness of the results.

math.DS↗

arXiv2025

Dynamics of Enveloping Semigroup of Flows

In this article, we relate the dynamics of a flow $(X, T)$ with the dynamics of the induced flow $(E(X), T)$ where $E(X)$ is the enveloping semigroup of flow $(X, T)$. We establish that a flow $(X, T)$ is distal if and only if the induced flow is also distal. We prove that while the induced flow on the enveloping semigroup of an equicontinuous flow is equicontinuous, the converse holds when $(X, T)$ is point transitive. We prove that for any flow $(X,T)$, $(E(X),T)$ is isomorphic to $(E(E(X)),T)$. We show that if a distal flow contains a minimal sensitive subsystem, the induced flow on the enveloping semigroup is also sensitive. We relate various forms of rigidity for a flow with rigidity for the induced flow. We also establish that for any proximal flow $(X, T)$, if T is abelian then the induced flow $(E(X),T)$ is also proximal.

math.DS↗

arXiv2023

On Equicontinuity and Related Notions in Nonautonomous Dynamical Systems

In this work, we investigate the dynamics of a general non-autonomous system generated by a commutative family of homeomorphisms. In particular, we investigate properties such as periodicity, equicontinuity, minimality and transitivity for a general non-autonomous dynamical system. In \cite{sk2}, the authors derive necessary and sufficient conditions for a system to be minimal. We claim the result to be false and provide an example in support of our claim. Further, we correct the result to derive necessary and sufficient conditions for a non-autonomous system to be minimal. We prove that for an equicontinuous flow generated by a commutative family, while the system need not exhibit almost periodic points, if $x$ is almost periodic then every point in $\overline{\mathcal{O}_H(x)}$ is almost periodic. We further prove that in such a case, the set $\overline{\mathcal{O}_H(x)}$ is uniformly almost periodic and hence provide an analogous extension to a result known for the autonomous systems. We prove that a system generated by a commutative family is transitive if and only if it exhibits a point with dense orbit. We also prove that any minimal system generated by commutative family is either equicontinuous or has a dense set of sensitive points.

math.DS↗