Dynamics of number expansions and translation surfaces

This thesis uses dynamical systems to study two types of objects: number expansions (Chapters 1 and 2) and translation surfaces (Chapter 3). Our first chapter builds a broad, unifying theory for a large class of continued fraction algorithms producing what we call contracted Farey expansions. These algorithms are based on three ideas: (i) contraction of generalised continued fractions, (ii) induced transformations, and (iii) the natural extension of the Farey tent map. Within this theory, we find several well-studied algorithms; a new subfamily of superoptimal continued fractions with arbitrarily good convergence and approximation properties; and a unifying framework to prove several old and new results in Diophantine approximation. Chapter 2 introduces a new, one-parameter family of functions called skewed symmetric golden maps. Using tools from ergodic theory, we study the relative frequencies of digits typically occurring in number expansions produced by these maps. The central tool for our analysis is a mysterious phenomenon of our functions called matching, which has been recently observed and exploited to understand several other families of functions generating number expansions. Our final chapter deals with translation surfaces, i.e., surfaces obtained by gluing pairs of parallel, equal-length, and oppositely oriented edges of planar polygons. The Veech group of a translation surface is the group of Jacobians of its orientation-preserving affine automorphisms. We develop a novel algorithm to construct translation surfaces with prescribed lattice Veech groups in given strata. Our ideas are also used to obtain obstructions for the realisability of certain Veech groups in certain strata. In particular, we show that the square torus is the only translation surface in any minimal stratum whose Veech group is the modular group.