Aperiodic Sequences and Aperiodic Geodesics

TL;DR

This paper introduces a quantitative measure of aperiodicity in dynamical systems and demonstrates the existence of highly aperiodic sequences and geodesics in specific mathematical settings.

Contribution

It defines a new quantitative condition for aperiodicity and proves the existence of maximally aperiodic sequences and geodesics under this condition.

Findings

Existence of maximally aperiodic sequences in Bernoulli-shift

Existence of maximally aperiodic geodesics on hyperbolic manifolds

Quantitative measure effectively captures aperiodicity

Abstract

We introduce a quantitative condition on orbits of dynamical systems which measures their aperiodicity. We show the existence of sequences in the Bernoulli-shift and geodesics on closed hyperbolic manifolds which are as aperiodic as possible with respect to this condition.