math.DS

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# Title: 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 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 $\Omega$.
 Subjects: Dynamical Systems (math.DS) MSC classes: 37E45 Cite as: arXiv:2009.03111 [math.DS] (or arXiv:2009.03111v1 [math.DS] for this version)

## Submission history

From: José Aliste-Prieto [view email]
[v1] Mon, 7 Sep 2020 14:00:24 GMT (31kb)

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