TY - JOUR

T1 - Rotation Numbers and Rotation Classes on One-Dimensional Tiling Spaces

AU - Aliste-Prieto, José

AU - Rand, Betseygail

AU - Sadun, Lorenzo

N1 - Funding Information:
J.A.-P. acknowledges financial support from CONICYT FONDECYT REGULAR 1160975, CHILE. The authors thank Michael Baake, Alex Clark, Franz Gähler, Antoine Julien, Johannes Kellendonk, John Hunton, Jamie Walton, and Dan Rust for helpful discussions.
Publisher Copyright:
© 2021, Springer Nature Switzerland AG.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021

Y1 - 2021

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=85099909863&partnerID=8YFLogxK

U2 - 10.1007/s00023-021-01019-2

DO - 10.1007/s00023-021-01019-2

M3 - Article

AN - SCOPUS:85099909863

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0637

ER -