Research
Research Interests
Translation Surfaces
A translation surface can be thought of either as a collection of polygons with sides identified in parallel opposite pairs, or as a Riemann surface along with a holomorphic one-form. Translation surfaces are locally Euclidean with finitely many cone points. I am interested in combinatorial and probabilistic problems about families of trajectories on translation surfaces. For example, how random are the directions of saddle connections, trajectories that connect cone points? What can we say about the language complexity of billiard trajectories?
Dilation Surfaces
Recently, I've been thinking a lot about dilation surfaces, which are similar to translation surfaces. One way to think about them is as a collection of polygons with sides glued together in parallel opposite pairs by maps that are the composition of dilations by positive real factors and translations. I am interested many things about dilation surfaces, including their Veech groups (groups of symmetries), moduli spaces, and the dynamics of flows and foliations on these surfaces. I am also interested in the interplay between dilation surfaces, twisted measured laminations, and Teichmuller geometry, as well as related objects such as affine interval exchange transformations.
Papers
Below you will find my recent papers. For a brief introduction to any paper, please click on the down arrow next to the paper title.
Moduli spaces of complex affine and dilation surfaces, with Paul Apisa and Matt Bainbridge. Crelle's Journal (2023).
A complex affine surface can be thought of as an atlas of charts on a Riemann surface where the transition maps are complex affine. We also allow these surfaces to have finitely many cone points. Dilation surfaces are then complex affine surfaces whose transition maps consist of real dilation and complex translation.
We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech, we show that the the moduli space of affine surfaces with fixed genus and with cone points of fixed complex order is a holomorphic affine bundle over the moduli space of Riemann surfaces. Similarly, the moduli space of dilation surfaces is a covering space of the moduli space of Riemann surfaces. We classify the connected components of the moduli space of dilation surfaces and show that any component is an orbifold K(G,1) where G is the framed mapping class group of Calderon-Salter.
Slope gap distributions of Veech surfaces, with Luis Kumanduri and Anthony Sanchez. To appear in Algebraic & Geometric Topology.
Slope gap distributions of translation surfaces are a way of gauging how random the directions of saddle connections on a translation surface are. It is known that Veech surfaces have slope gap distributions that are piecewise real analytic. This result is obtained by giving an algorithm to compute the slope gap distributions of Veech surfaces and analyzing the output.
In this paper, we modify the algorithm and use our modification to show that not only are the slope gap distributions of Veech surfaces piecewise real analytic, but also that they necessarily have only finitely many points of non-analyticity. As a result of this modification, we also show that the slope gap distributions of Veech surfaces have quadratic tail decay.
A one-holed dilation torus is a genus one dilation surface with a single boundary component. They are simple examples of dilation surfaces on which we can understand the dynamics quite well. An open question in the study of dilation surfaces is to determine the typical dynamical behavior of the directional flow on a fixed dilation surface. In this paper, we build on works by other authors to carefully understand the directional flow on one-holed dilation tori.
We show that on any one-holed dilation torus, in all but a measure zero Cantor set of directions, the directional flow has an attracting periodic orbit, is minimal, or is completely periodic. We further show using new methods that in this Cantor set of directions, the directional flow is attracted to either a saddle connection or a lamination on the surface that is locally the product of a measure zero Cantor set and an interval.
Dilation surfaces are a generalization of translation surfaces. One way to think about dilation surfaces is as a collection of polygons in the plane with sides identified in parallel opposite pairs by dilation and translation, versus just by translation for translation surfaces. While much is known about the geometry and dynamics of translation surfaces and their moduli spaces, much less is known about dilation surfaces.
This paper focuses on the realization problem, which asks: what topological maps (mapping class group elements) can be realized as geometric maps (affine automorphisms) on some dilation surface? After surveying the landscape of this problem for translation surfaces, we discuss new features of dilation surfaces called exotic Dehn twists that make solving this problem more difficult for dilation surfaces. Nevertheless, we will establish that only certain types of mapping class group elements can be realized in the affine automorphism groups of dilation surfaces. We will also present a generalization of Thurston's construction that constructs from a pair of filling multicurves a dilation surface with a pair of Dehn multitwists in its affine automorphism group.
W-measurable sensitivity of semigroup actions, with F. Bozgan, A. Sanchez, J. Spielberg, D. Stevens, and C. E. Silva. Colloquium Mathematicum (2021).
In the theory of transformations on metric spaces, a transformation is said to have sensitive dependence on initial conditions if small differences in the position of a point originally can yield large differences after iterations of the transformation. There have been various generalizations of the idea of sensitive dependence to the context of measurable transformations, including W-measurable sensitivity. There have also been generalizations in topological dynamics to group actions.
In this paper, we begin to bridge the gap between these two directions of generalization to consider W-measurable sensitivity of a class of semigroup actions. We prove a rigidity theorem that roughly states that under certain initial conditions, these semigroup actions are either W-measurably sensitive or isomorphic modulo a measure zero set to an action by isometries. We also show that W-measurable sensitivity is not preserved under factors, but is preserved when restricted to suitably large sub-semigroups (e.g. syndetic or thick).
Statistics of square-tiled surfaces: symmetry and short loops, with Sunrose Shretha. J. Experimenal Mathematics (2020).
A square-tiled surface is a special type of translation surface that is constructed by gluing together finitely many unit squares. Because the relative periods of such surfaces are always integral, square-tiled surfaces are often thought of as the integral points in the strata of translation surfaces. Counting square tiled surfaces is then related to calculating the Masur-Veech volumes of such strata.
The simplest square-tiled surface is the one-square torus, which has many nice symmetry properties involving its holonomy vectors and Veech group. In this paper, we consider families of square-tiled surfaces that share properties with the square torus. We prove relationships between these properties, and show that some of these properties are rare by proving that the number of square-tiled surfaces with these properties are finite in each stratum.
Twisted quadratic differentials, also known as dilation surfaces, are geometric structures that are in a way a generalization of translation surfaces. One way to define a dilation surface is as a collection of polygons with sides identified by translations and dilations by nonzero real factors, whereas translation surfaces only allow side identifications by translations. Just with this small generalization, we get a new class of surfaces that exhibit interesting new dynamical behaviors.
In this thesis, we start by giving a thorough introduction to dilation surfaces, including a study of the moduli space of genus one dilation surfaces. We then move on to tackling the realization problem for dilation surface. That is, we ask what elements and subsets of the mapping class group can be realized in the affine automorphism group of some dilation surface. We demonstrate how to construct dilation surfaces with a given pseudo-Anosov map in their affine automorphism group, show the existence of exotic Dehn twists, and construct dilation surfaces with simultaneous Dehn multitwists. The last construction gives rise to dilation surfaces with large affine automorphism groups.
Subsequence rational ergodicity of rank-one transformations, with F. Bozgan, A. Sanchez, D. Stevens, and C. E. Silva. J. Dynamical Systems (2015).
By the Birkhoff ergodic theorem, any ergodic transformation on a probability space satisfies the property that "time averages equal space averages" for almost every point. This no longer holds in the case of infinite measure systems as the time averages (the proportion of visits of the orbit of a point to a set) will converge to zero, and not the space average (the measure of the set). There are various modifications of the notion of convergence of ergodic averages to the setting of infinite measure systems, including weak rational ergodicity.
We work with a class of transformations called rank-one transformations. These are known to be generic in the space of measure-preserving transformations in both finite and infinite measure. We show that all rank-one transformations are weakly rationally ergodic along a subsequence, but that there exist rank-one transformations that are not weakly rationally ergodic. We also extend these results to the stronger notion of bounded rational ergodicity.
In the context of Ergodic Theory, for every weak mixing N-action, there is a density one subset of N for which the system is then strong mixing. In this sense, every weak mixing N-action is almost strong mixing.
We ask then if a similar statement holds for topological dynamical systems. We show that under suitable initial conditions, in the context of topological dynamics, a weak mixing group action by an abelian group is always almost strong mixing in the sense that it is strong mixing on a thick subset of the group. To get there, we first prove that a system being almost strong mixing is equivalent to it being k-transitive for all k.