Rotation numbers and rotation classes on one-dimensional 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 Poncaré'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 $\Omega$.

References in corpus (1)

• Self affine Delone sets and deviation phenomena