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Title: Towards a classication of constant length substitution tiling spaces
Authors: Al-Hindawe, Ahmed L. M.
Supervisors: Clark, Alex
Neumann, Frank
Award date: 15-Mar-2019
Presented at: University of Leicester
Abstract: We focus on families of hyperbolic attractors for diffeomorphisms of the solid surface of genus two which admit the structure of a substitution tiling space for a substitution of constant length. These spaces are similar to solenoids in that each one admits an almost one to one map onto a solenoid. In particular, we investigate a special class we call difference d substitutions. The simplest families, where d = 1, are very close to solenoids in that they can be considered as a solenoid with one path component replaced with a pair of asymptotic path components. Despite this close link to the underlying solenoid, their topological classification is considerably more complicated than those for solenoids, and there is more than one homeomorphism class of difference 1 substitution tiling space corresponding to the same n-adic solenoid. In particular, we show that two difference one constant length substitution tiling spaces arising from the substitutions θ and θ’ are homeomorphic if and only if the shift on the subshift Ω is topologically conjugate to the shift on Ω0 or the inverse of the shift on Ω0 . Using the known classification of these shifts based on [15], this allows a complete classification of these families. We then carry out analysis with the general and more complex case where d > 1 . The three key elements in our analysis are a detailed analysis of a factor map onto a solenoid based on a construction of Coven and Keane and the classification at which they arrive [15], the structure of affine maps of solenoids and the rigidity of substitution tiling spaces determined by Barge, Swanson [7] and Kwapisz [20].
Type: Thesis
Level: Doctoral
Qualification: PhD
Rights: Copyright © the author. All rights reserved.
Appears in Collections:Leicester Theses
Theses, Dept. of Mathematics

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