# Dynamical properties of random walks

## TL;DR

This paper investigates the dynamical behaviors such as hypercyclicity and chaos of transition operators linked to countable Markov chains, including simple random walks on integers and positive integers.

## Contribution

It introduces a detailed analysis of dynamical properties of transition operators for Markov chains, extending understanding of their chaotic and hypercyclic behaviors.

## Key findings

- Transition operators exhibit hypercyclicity and chaos under certain conditions.
- Simple random walks on Z and Z+ demonstrate specific dynamical properties.
- The study provides criteria for dynamical behaviors in Markov chain operators.

## Abstract

In this paper, we study dynamical properties as hypercyclicity, supercyclicity, frequent hypercyclicity and chaoticity for transition operators associated to countable irreductible Markov chains. As particular cases, we consider simple random walks on Z and Z+.