A Characterization of Rotation Number on One-Dimensional Tiling Spaces

TL;DR

This paper extends the concept of rotation numbers to non-periodic 1-dimensional tiling spaces, establishing conditions for their well-definedness and illustrating cases where these conditions fail.

Contribution

It introduces a new framework for rotation numbers in tiling spaces and identifies necessary assumptions for their global well-definedness.

Findings

Conditions for well-defined rotation numbers are sufficient.

Counterexamples show failure cases without these conditions.

Provides a generalization of classical rotation number theory.

Abstract

Identity-homotopic self-homeomorphisms of a space of non-periodic 1-dimensional tiling are generalizations of orientation-preserving self-homeomorphisms of circles. We define the analogue of rotation numbers for such maps. In constrast to the classical situation, additional assumptions are required to make rotation numbers globally well-defined and independent of initial conditions. We prove that these conditions are sufficient, and provide counterexamples when these conditions are not met.