# Linear Dynamics Induced by Odometers

## TL;DR

This paper investigates the dynamical properties of composition operators induced by measures on odometers, focusing on their ability to generate mixing and transitive linear operators on $L^p$ spaces, expanding the understanding of linear dynamics.

## Contribution

It introduces and analyzes a new class of composition operators on $L^p$ spaces derived from measures on odometers, highlighting their mixing and transitivity properties.

## Key findings

- Measures on odometers can induce mixing operators.
- Measures on odometers can induce transitive operators.
- The study broadens the class of concrete operators in linear dynamics.

## Abstract

Weighted shifts are an important concrete class of operators in linear dynamics. In particular, they are an essential tool in distinguishing variety dynamical properties. Recently, a systematic study of dynamical properties of composition operators on $L^p$ spaces has been initiated. This class of operators includes weighted shifts and also allows flexibility in construction of other concrete examples. In this article, we study one such concrete class of operators, namely composition operators induced by measures on odometers. In particular, we study measures on odometers which induce mixing and transitive linear operators on $L^p$ spaces.