A CHARACTERIZATION OF ROTATION NUMBER ON ONE-DIMENSIONAL TILING SPACES BETSEYGAIL RAND AND LORENZO SADUN Abstract. Identity-homotopic self-homeomorphisms of a space of non- periodic 1-dimensional tiling are generalizations of orientation-preserving self-homeomorphisms of circles. We define the analogue of rotation num- bers for such maps. In constrast to the classical situation, additional as- sumptions are required to make rotation numbers globally well-defined and independent of initial conditions. We prove that these conditions are sufficient, and provide counterexamples when these conditions are not met. 1. Introduction Since 1885, rotations numbers have been used to understand orientation- preserving homemorphisms of the circle S 1 = R/Z. Each such homeomor- phism f lifts to a map F : R → R such that F (x+1) = F (x)+1 for all x ∈ R. The rotation number of f (see e.g. [3]) is defined to be lim n→∞ F n (x)−x n . Dif- ferent lifts give rotation numbers that differ by integers, so we should view the rotation number as an element of R/Z. Viewed in this way, the rotation number is always well-defined and does not depend on the initial point x. If L is any length, the circle R/LZ can be viewed as the orbit, under translations, of a tiling that is periodic with period L [5]. We can then view an orientation-preserving homeomorphism of R/LZ as a map on a space of periodic tilings. The goal of this paper is to define and study an analogue of rotation numbers for self-homeomorphisms, homotopic to the identity (“identity-homotopic”) of a space of non-periodic tilings of R. We restrict our attention to non-periodic tilings that have two simplifying properties: • Finite local complexity (FLC): For each radius R, there are only finitely many patterns of size R or smaller, up to translation. Under the topology described below, this is equivalent to the tiling space being compact. • Repetitivity: For each pattern P that appears in our given tiling T , there exists a radius R such that every ball of radius R in T contains at least one occurrence of P . This is equivalent to the orbit closure of T being a minimal dynamical system. That is, if Ω is the orbit 1 arXiv:1708.00901v1 [math.DS] 2 Aug 2017