Finiteness conditions on 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.

We classify the locally finite ergodic invariant measures of certain infinite interval exchange transformations (IETs). These transformations naturally arise from return maps of the straight-line flow on certain translation surfaces, and the study of the invariant measures for these IETs is equivalent to the study of invariant measures for the straight-line flow in some direction on these translation surfaces. For the surfaces and directions for which our methods apply, we can characterize the locally finite ergodic invariant measures of the straight-line flow in a set of directions of Hausdorff dimension larger than 1/2. We promote this characterization to a classification in some cases. For instance, when the surfaces admit a cocompact action by a nilpotent group, we prove each ergodic invariant measure for the straight-line flow is a Maharam measure, and we describe precisely which Maharam measures arise. When the surfaces under consideration are finite area, the straight-line flows in the directions we understand are uniquely ergodic. Our methods apply to translation surfaces admitting multi-twists in a pair of cylinder decompositions in non-parallel directions.

The Veech group of a translation surface is the group of Jacobians of orientation-preserving affine automorphisms of the surface. We present an algorithm which constructs all translation surfaces with a given lattice Veech group in any given stratum. In developing this algorithm, we give a new proof of a finiteness result of Smillie and Weiss, namely that there are only finitely many unit-area translation surfaces in any stratum with the same lattice Veech group. Our methods can be applied to obtain obstructions of lattices being realized as 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 all of textrm{SL}_2mathbb {Z}.

In translation surfaces of finite area (corresponding to holomorphic differentials), directions of saddle connections are dense in the unit circle. On the contrary, saddle connections are fewer in translation surfaces with poles (corresponding to meromorphic differentials). The Cantor-Bendixson rank of their set of directions is a measure of descriptive set-theoretic complexity. Drawing on a previous work of David Aulicino, we prove a sharp upper bound that depends only on the genus of the underlying topological surface. The proof uses a new geometric lemma stating that in a sequence of three nested invariant subsurfaces the genus of the third one is always bigger than the genus of the first one.

We study the boundary of an affine invariant submanifold of a stratum of translation surfaces in a partial compactification consisting of all finite area Abelian differentials over nodal Riemann surfaces, modulo zero area components. The main result is a formula for the tangent space to the boundary. We also prove finiteness results concerning cylinders, a partial converse to the Cylinder Deformation Theorem, and a result generalizing part of the Veech dichotomy.

We consider a symbolic coding of linear trajectories in the regular octagon with opposite sides identified (and more generally in regular 2n-gons). Each infinite trajectory gives a cutting sequence corresponding to the sequence of sides hit. We give an explicit characterization of these cutting sequences. The cutting sequences for the square are the well-studied Sturmian sequences which can be analysed in terms of the continued fraction expansion of the slope. We introduce an analogous continued fraction algorithm which we use to connect the cutting sequence of a trajectory with its slope. Our continued fraction expansion of the slope gives an explicit sequence of substitution operations which generate the cutting sequences of trajectories with that slope. Our algorithm can be understood in terms of renormalization of the octagon translation surface by elements of the Veech group.