# Escaping a neighborhood along a prescribed sequence in Lie groups and Banach algebras

## TL;DR

This paper explores sequences that describe how elements in Lie groups and Banach algebras can escape neighborhoods of the identity, revealing their spectral properties and connections to linear dynamics.

## Contribution

It introduces new spectral characterizations of Jamison sequences and links their properties to the behavior of elements in Lie groups and Banach algebras.

## Key findings

- Jamison sequences characterize elements escaping neighborhoods in Lie groups and Banach algebras.
- Spectral properties of Jamison sequences are established.
- Connections to linear dynamics and operator theory are demonstrated.

## Abstract

It is shown that Jamison sequences, introduced in 2007 by Badea and Grivaux ([C. Badea and S. Grivaux, Unimodular eigenvalues, uniformly distributed sequences and linear dynamics, Adv. Math. 211 (2007), no. 2, 766--793]), arise naturally in the study of topological groups with no small subgroups, of Banach or normed algebra elements whose powers are close to identity along subsequences, and in characterizations of (self-adjoint) positive operators by the accretiveness of some of their powers. The common core of these results is a description of those sequences for which non-identity elements in Lie groups or normed algebras escape an arbitrary small neighborhood of the identity in a number of steps belonging to the given sequence. Several spectral characterizations of Jamison sequences are given and other related results are proved.