Topological properties of tiles and digit sets
LE3 .A278 2009
2009
Curry, Eva
Acadia University
Master of Science
Masters
Mathematics
Mathematics & Statistics
This thesis investigates topological properties of tiles, T(A;D), arising from iterated function systems generated by dilation matrices, A, and associated digit sets, D. We develop a new method for investigating connectedness of T(A;D) via level sets of the digit set and matrix similarity. We conjecture that there exists digit set DA for any dilation matrix A, for which T(A;DA) is connected. We prove this conjecture for several important cases. While it has previously been shown in the two dimensional case that digit sets exist for which T(A;D) is connected [5], we provide a new, constructive method that we expect will easily generalize to higher dimensions, where no results of this form are currently known. We also develop new computational tools for investigating tiles T(A;D), including an implementation of the neighbour-nding algorithm of Scheicher and Thuswaldner [20].
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https://scholar.acadiau.ca/islandora/object/theses:137