Rotation numbers and rotation classes on one-dimensional tiling spaces

TL;DR

This paper generalizes rotation theory from circle maps to one-dimensional tiling spaces, introducing a cohomology class as an analogue of the rotation number and establishing key existence, uniqueness, and conjugacy results.

Contribution

It develops a cohomological framework for rotation numbers in tiling spaces, extending classical results and establishing conditions for semi-conjugacy to translation.

Findings

Defined a cohomology class as a rotation number analogue

Proved existence and uniqueness of the cohomology class

Established an analogue of Poincaré's theorem for tiling spaces

Abstract

We extend rotation theory of circle maps to tiling spaces. Specifically, we consider a 1-dimensional tiling space $\Omega$ with finite local complexity and study self-maps $F$ that are homotopic to the identity and whose displacements are strongly pattern equivariant (sPE). In place of the familiar rotation number we define a cohomology class $[\mu]$. We prove existence and uniqueness results for this class, develop a notion of irrationality, and prove an analogue of Poncar\'{e}'s Theorem: If $[\mu]$ is irrational, then $F$ is semi-conjugate to uniform translation on a space $\Omega_\mu$ of tilings that is homeomorphic to $\Omega$. In such cases, $F$ is semi-conjugate to uniform translation on $\Omega$ itself if and only if $[\mu]$ lies in a certain subspace of the first cohomology group of $\Omega$.