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Topologically Mixing Tilings of R^2 Generated by a Generalized Substitution Open Access

This dissertation presents sufficient conditions for a substitution tiling dynamical system of R2 generated by a generalized substitution on 3 letters to be topologically mixing. These conditions are shown to hold on a large class of tiling substitutions originally presented by Kenyon in 1996. This problem was suggested by Boris Solomyak, and many of the techniques that are used in this dissertation are based on the work by Kenyon, Sadun, and Solomyak in 2005. They studied one-dimensional tiling dynamical systems generated on 2 letters and provided similar conditions that were sufficient to ensure the one-dimensional tiling substitution was topologically mixing.If a tiling dynamical system of R2 satisfies our conditions (and thus is topologically mixing), then we can construct additional topologically mixing tiling dynamical systems of R2. By considering the stepped surface constructed by a tiling Tσ, we can get a new tiling of R2 by projecting the surface orthogonally onto an irrational plane through the origin. If the projection is the Perron-Frobenius projection, then the substitution tiling dynamical system is pseudo-self-similar.

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