arXiv subjects
Publications and source records attributed to Nayan Adhikary.
The dual concepts of `universality' and `hypercyclicity' are better understood and studied as `topological transitivity'. In this article we consider transitivity properties of skew products, essentially with non-compact fibers. We study the `Universality Conditions' and `Hypercyclicity Criterion' associated with the dynamical properties of transitivity, weakly mixing and mixing for these skew products.
In this article we give several characterizations for various transitivity properties for linear operators. We define a general form of `Hypercyclicity Criterion' using a Furstenberg family $\mathcal{F}$ to characterize $\mathcal{F}$-transitive operators. In particular, we find an equivalent characterization for mixing operators. We study proximal and asymptotic relations for linear operators and prove that the difference between mixing operators and Kitai's Criterion can be presented through these relations. Finally, we find an equivalent characterization of strongly transitive abd strongly product transitive operators.
We consider the topological dynamics of closed relations(CR) by studying one of the oldest dynamical property - `transitivity'. We investigate the two kinds of (closed relation) CR-dynamical systems - $(X,G)$ where the relation $G \subseteq X \times X$ is closed and $(X,G, \bullet)$ giving the `suitable dynamics' for a suitable closed relation $G$, where $X$ is assumed to be a compact metric space without isolated points. $(X,G)$ gives a general approach to study initial value problems for a set of initial conditions, whereas $(X,G, \bullet)$ gives a general approach to study the dynamics of both continuous and quasi-continuous maps. We observe that the dynamics of closed relations is richer than the dynamics of maps and find that we have much more versions of transitivity for these closed relations than what is known for maps.
The underlying theme of this article is a class of sequences in metric structures satisfying a much weaker kind of Cauchy condition, namely quasi-Cauchy sequences (introduced in \cite{bc}) that has been used to define several new concepts in recent articles \cite{PDSPNA2, PDSPNA1}. We first consider a weaker notion of precompactness based on the idea of quasi-Cauchy sequences and establish several results including a new characterization of compactness in metric spaces. Next we consider associated idea of continuity, namely, ward continuous functions \cite{caka}, as this class of functions strictly lies between the classes of continuous and uniformly continuous functions and mainly establish certain coincidence results. Finally a new class of Lipschitz functions called "quasi-Cauchy Lipschitz functions" is introduced following the line of investigations in \cite{Beer1,Beer2,Beer3,g1} and again several coincidence results are proved. The motivation behind such kind of Lipschitz functions is ascertained by the observation that every real valued ward continuous function defined on a metric space can be uniformly approximated by real valued quasi-Cauchy Lipschitz functions.