Minoration of the complexity function associated to a translation on the torus

TL;DR

This paper establishes a lower bound on the complexity function of piecewise translation maps conjugated to minimal translations on the torus, showing it grows at least linearly with respect to the translation dimension.

Contribution

It provides a new lower bound for the complexity function of certain dynamical systems on the torus, extending understanding of their combinatorial complexity.

Findings

The complexity function p_k(n) is at least kn+1 for all integers n.

The result applies to piecewise translation maps conjugated to minimal torus translations.

It advances the theoretical understanding of the combinatorial complexity in toral dynamical systems.

Abstract

We show that the complexity function $p_k(n)$ of a piecewise translation map conjugated to a minimal translation on the torus $\TT^k = \RR^k / \ZZ^k$ is at least $kn+1$ for every integer $n$.