Homeomorphisms of One-dimensional Inverse Limits with Applications to Substitution Tilings, Unstable Manifolds, and Tent Maps

1999/05/31 by Marcy Barge, Barge, Marcy, James Jacklitch +3
Materials Science · Mathematics · #54F15 #54H20 #58F03 #58F12 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals #Mathematics and Applications #Quasicrystal Structures and Properties #math.DS #math.GT #msc:54F15 #msc:54H20 #msc:58F03 #msc:58F12

paper · pdf · doi:10.48550/arxiv.math/9905197

14 pages

arxiv created 1999/05/31 · openalex publication_date 1999/05/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that f and g are Markov surjections, each defined on a wedge of circles, each fixing the branch point and having the branch point as the only critical value. We show that if the points in the inverse limit spaces associated with f and g corresponding to the branch point are distinguished then these inverse limit spaces are homeomorphic if and only if the substitutions associated with f and g are weakly equivalent. This, and related results, are applied to one-dimensional substitution tiling spaces, one-dimensional unstable manifolds of hyperbolic sets, and inverse limits of tent maps with periodic critical points.

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