We investigate two different topics in discrete mathematics: the geometry of piecewise-linear surfaces and the long-term behavior of discrete dynamical systems. A regular polygon surface M is a surface graph (Sigma, Gamma) together with a continuous map psi from Sigma into Euclidean 3-space which maps faces to regular Euclidean polygons. When Sigma is homeomorphic to the sphere, and the degree of every face of Gamma is five, we prove that M can be realized as the boundary of a union of dodecahedra glued together along common facets. Under the same assumptions but when the faces of Gamma have degree four or eight, we prove that M can be realized as the boundary of a union of cubes and octagonal prisms glued together along common facets. We exhibit counterexamples showing the failure of both theorems for higher genus surfaces. In joint work with Richard Kenyon and Ren Yi we study domain exchange maps (DEMs). A DEM is a dynamical system defined on a smooth Jordan domain which is a piecewise translation. We explain how to use cut-and-project sets to construct minimal DEMs. Specializing to the case in which the domain is a square and the cut-and-project set is associated to a Galois lattice, we construct an infinite family of DEMs in which each map is associated to a PV number. We develop a renormalization scheme for these DEMs. Certain DEMs in the family can be composed to create multistage, renormalizable DEMs.
Alevy, Ian,
"Regular Polygon Surfaces and Renormalizable Rectangle Exchange Maps"
(2018).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/nnms-xe64
