A Characterization of Rotation Number on One-Dimensional Tiling Spaces

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.

Many of the results in this paper had previously appeared in papers by A. Clark, by Y. Shvestov, and by J. Aliste-Prieto