# Pointwise dynamics under Orbital Convergence

## TL;DR

This paper establishes conditions for the preservation of dynamical properties at points under various types of convergence and provides examples illustrating the topological nature of these properties.

## Contribution

It introduces sufficient conditions for pointwise dynamical behaviors to be preserved under convergence and analyzes the topological structure of sets of points with specific properties.

## Key findings

- Sets of expansive, positively expansive, and sensitive points are neither open nor closed.
- Sets of transitive and mixing points are closed but not open.
- Properties like expansivity and sensitivity are not necessarily preserved under uniform convergence.

## Abstract

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 $\alpha$-persistence at a point need not be preserved under pointwise convergence.