# Recurrent strongly continuous operator families on Banach space

## TL;DR

This paper studies recurrent and rigid $C_0$-semigroups on Banach spaces, providing structural characterizations and conditions for chaos, especially in translation semigroups on weighted function spaces.

## Contribution

It introduces the concept of (uniformly) $C_0$-rigid semigroups and characterizes recurrent $C_0$-semigroups induced by semiflows.

## Key findings

- Characterization of Li-Yorke chaos in translation semigroups
- Structural description of $C_0$-rigid semigroups
- Conditions for recurrence in semigroups induced by semiflows

## Abstract

In this paper, we analyze recurrent $C_{0}$-semigroups of bounded operators on Banach spaces. We also introduce the notion of a (uniformly) $C_{0}$-rigid semigroups of bounded operators and give a structural characterization of them. A characterization of a Li-Yorke chaoticity of the translation semigroup $(T(t))_{t\geq 0}$ on weighted spaces of integrable functions and continuous functions in terms of admissible weight function is given. The recurrent $C_0$-semigroups induced by semiflows are characterized on the spaces of integrable functions and of spaces of continuous functions.

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Source: https://tomesphere.com/paper/1905.00407