Dynamics of the semigroup of contractive automorphisms of Banach spaces

TL;DR

This paper investigates the dynamic behavior of contractive automorphism semigroups in Banach spaces, emphasizing their metric properties, orbit sizes, and influence on the space's geometry, especially in finite dimensions.

Contribution

It introduces a detailed analysis of the metric dynamics of automorphism semigroups and explores their geometric implications in finite-dimensional Banach spaces.

Findings

Characterization of orbit sizes under automorphism semigroups

Insights into semitransitivity properties of these semigroups

Impact on the geometry of the unit ball in finite-dimensional spaces

Abstract

Motivated by some recent twaddles on Mazur rotations problem, we study the "dynamics" of the semigroup of contractive automorphisms of Banach spaces, mostly in finite-dimensional spaces. We focus on the metric aspects of the "action" of such semigroups, the size of the orbits and semitransitivity properties, and their impact on the geometry of the unit ball of the underlying space.