Ruchi Das

Expansive self-homeomorphisms on G-spaces

Beginning with examples, the notion ofG-expansiveness over a metric spaceX on which a topological groupG acts is introduced. Some conditions are determined under which expansiveness onX impliesG-expansiveness. A characterization of aG-expansive homeomorphism is obtained which in turn gives a sufficient condition for the homeomorphic extension of aG-expansive homeomorphism to beG-expansive. At the end, some results are stated in the form of concluding remarks.

A note on representation of pseudovariant maps

We define and study the notions of pseudovariant maps and pseudogeny maps on G-spaces. We prove that the set of all pseudogenies of a locally comapct G-space is a group. We further obtain representation of pseudovariant maps in terms of pseudogenies. Finally, we obtain Tietze type extension result for pseudovariant homotopies defined on locally compact second countable G-spaces.

On Nonautonomous Discrete Dynamical Systems

We define and study expansiveness, shadowing, and topological stability for a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a metric space.

Spectral decomposition theorem in equicontinuous nonautonomous discrete dynamical systems

In this paper, we define the notion of weak chain recurrence and study properties of weak chain recurrent sets in a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a compact metric space. Our main result is the Smale’s spectral decomposition theorem in an equicontinuous nonautonomous discrete dynamical system.

Some properties of chain recurrent sets in a nonautonomous discrete dynamical system

Abstract In this paper, we define chain recurrence and study properties of chain recurrent sets in a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a compact metric space. We also study chain recurrent sets in a nonautonomous discrete system having shadowing property.

On properties of G-expansive homeomorphisms

We show that LUB of the set of G-expansive constants for a G-expansive homeomorphism h on a compact metric G-space, G compact, is not a G-expansive constant for h. We obtain a result regarding projecting and lifting of G-expansive homeomorphisms having interesting applications. We also prove that the G-expansiveness is a dynamical property for homeomorphisms on compact metric G-spaces and study G-periodic points.

Transitivities of maps on G-spaces

Abstract In this paper, we define various kinds of transitivity of maps on G-spaces. We obtain conditions on G-spaces and on maps for one type of transitivity to imply another type of transitivity. Giving several examples and proving various equivalences, we provide a complete description of the relationships among the different types of transitivities defined for maps on G-spaces.

Asymptotic Properties of -Expansive Homeomorphisms on a Metric -Space

We define and study the notions of positively and negatively -asymptotic points for a homeomorphism on a metric -space. We obtain necessary and sufficient conditions for two points to be positively/negatively -asymptotic. Also, we show that the problem of studying -expansive homeomorphisms on a bounded subset of a normed linear -space is equivalent to the problem of studying linear -expansive homeomorphisms on a bounded subset of another normed linear -space.

Sensitivity, property P and uniform entropy

We give a sufficient condition for sensitivity of continuous maps defined on a uniform space. We also study property P for uniformly continuous maps defined on a uniform space and its relation with uniform entropy.

On collective sensitivity for -actions

ABSTRACT We define and study here the notion of k-type collective sensitivity and related notions for a -action on a metric space. A characterization of collective sensitivity for a -action on a compact metric space is obtained and its relation with k-type weak mixing is studied. We prove that k-type collective sensitivity is preserved under uniform conjugacy and finite product.