We extend rotation theory of circle maps to tiling spaces. Specifically, we consider a one-dimensional tiling space Ω with finite local complexity and study self-maps F that are homotopic to the identity and whose displacements are strongly pattern equivariant. In place of the familiar rotation number, we define a cohomology class [μ] ∈ Hˇ 1(Ω , R). We prove existence and uniqueness results for this class, develop a notion of irrationality, and prove an analogue of Poincaré’s theorem: If [μ] is irrational, then F is semi-conjugate to uniform translation on a space Ω μ of tilings that is homeomorphic to Ω. In such cases, F is semi-conjugate to uniform translation on Ω itself if and only if [μ] lies in a certain subspace of Hˇ 1(Ω , R).
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics