arXiv subjects

Jayadev Athreya

Publications and source records attributed to Jayadev Athreya.

10 recordsLinked to original sources

arXiv2025

Heisenberg Translation Flows

We study ergodic theoretical properties of flows on circle bundles over translation surfaces that arise via prequantization, generalizing the theory of Heisenberg nilflows to base surfaces more general than tori; these flows are among the most fundamental examples of parabolic dynamical systems with non-trivial central directions. In particular, we show that such flows are relatively mixing, i.e., they exhibit decay of correlations in the orthogonal complement of functions constant along fibers. We discuss applications of this result to the dynamics of such flows, to the ergodic theory on the corresponding space of wave functions, and, via surface of section constructions, to the study of affine skew products over interval exchange transformations, in the spirit of Furstenberg's classification program for measurable dynamical systems.

math.DS↗

arXiv2025

Semiclassical measures for complex hyperbolic quotients

We study semiclassical measures for Laplacian eigenfunctions on compact complex hyperbolic quotients. Geodesic flows on these quotients are a model case of hyperbolic dynamical systems with different expansion/contraction rates in different directions. We show that the support of any semiclassical measure is either equal to the entire cosphere bundle or contains the cosphere bundle of a compact immersed totally geodesic complex submanifold. The proof uses the one-dimensional fractal uncertainty principle of Bourgain-Dyatlov [arXiv:1612.09040] along the fast expanding/contracting directions, in a way similar to the work of Dyatlov-Jézéquel [arXiv:2108.10463] in the toy model of quantum cat maps, together with a description of the closures of fast unstable/stable trajectories relying on Ratner theory.

math.AP↗

arXiv2025

Complexity for billiards in regular N-gons

We compute the complexity of the billiard language of the regular Euclidean $N$-gons (and other families of rational lattice polygons), answering a question posed by Cassaigne-Hubert-Troubetzkoy. Our key technical result is a counting result for saddle connections on lattice surfaces, when we count by combinatorial length.

math.DS↗

arXiv2024

Spectral decomposition and Siegel-Veech transforms for strata: The case of marked tori

Generalizing the well-known construction of Eisenstein series on the modular curves, Siegel-Veech transforms provide a natural construction of square-integrable functions on strata of differentials on Riemannian surfaces. This space carries actions of the foliated Laplacian derived from the SL(2,R)-action as well as various differential operators related to relative period translations. In the paper we give spectral decompositions for the stratum of tori with two marked points. This is a homogeneous space for a special affine group, which is not reductive and thus does not fall into well-studied cases of the Langlands program, but still allows to employ techniques from representation theory and global analysis. Even for this simple stratum exhibiting all Siegel-Veech transforms requires novel configurations of saddle connections. We also show that the contiunuous spectrum of the foliated Laplacian is much larger than the space of Siegel-Veech transforms, as opposed to the case of the modular curve. This defect can be remedied by using instead a compound Laplacian involving relative period translations.

math.NT↗

arXiv2023

Currents with corners and counting weighted triangulations

Let $Σ$ be a closed orientable hyperbolic surface. We introduce the notion of a \textit{geodesic current with corners} on $Σ$, which behaves like a geodesic current away from certain singularities (the "corners"). We topologize the space of all currents with corners and study its properties. We prove that the space of currents with corners shares many properties with the space of geodesic currents, although crucially, there is no canonical action of the mapping class group nor is there a continuous intersection form. To circumvent these difficulties, we focus on those currents with corners arising from harmonic maps of graphs into $Σ$. This leads to the space of \textit{marked harmonic currents with corners}, which admits a natural Borel action by the mapping class group, and an analog of Bonahon's\cite{Bonahon} compactness criterion for sub-level sets of the intersection form against a filling current. As an application, we consider an analog of a curve counting problem on $Σ$ for triangulations. Fixing an embedding $ϕ$ of a weighted graph $Γ$ into $Σ$ whose image $ϕ(Γ)$ is a triangulation of $Σ$, let $N_ϕ(L)$ denote the number of mapping classes $f$ so that a weighted-length minimizing representative in the homotopy class determined by $f \circ ϕ$ has length at most $L$. In analogy with theorems of Mirzakhani\cite{Mirzakhani}, Erlandsson-Souto\cite{ErlandssonSouto}, and Rafi-Souto\cite{RafiSouto}, we prove that $N_ϕ(L)$ grows polynomially of degree $6g-6$ and the limit \[ \lim_{L \rightarrow \infty} \frac{N_ϕ(L)}{L^{6g-6}}\] exists and has an explicit interpretation depending on the geometry of $Σ$, the vector of weights, and the combinatorics of $ϕ$ and $Γ$.

math.GT↗

arXiv2022

Stable Random Fields, Patterson-Sullivan measures and Extremal Cocycle Growth

We study extreme values of group-indexed stable random fields for discrete groups $G$ acting geometrically on spaces $X$ in the following cases: 1) $G$ acts freely, properly discontinuously by isometries on a CAT(-1) space $X$, 2) $G$ is a lattice in a higher rank Lie group, acting on a symmetric space $X$, 3) $G$ is the mapping class group of a surface acting on its Teichmuller space. The connection between extreme values and the geometric action is mediated by the action of the group $G$ on its limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth which measures the distortion of measures on the boundary in comparison to the movement of points in the space $X$ and show that its non-vanishing is equivalent to finiteness of the Bowen-Margulis measure for the associated unit tangent bundle $U(X/G)$ provided $X/G$ has non-arithmetic length spectrum. As a consequence, we establish a dichotomy for the growth-rate of a partial maxima sequence of stationary symmetric $α$-stable ($0 < α< 2$) random fields indexed by groups acting on such spaces. We also establish analogous results for normal subgroups of free groups.

math.DS↗

arXiv2019

Protecting billiard balls from collisions

We present a game inspired by research on the possible number of billiard ball collisions in the whole Euclidean space. One player tries to place $n$ static "balls" with zero radius (i.e., points) in a way that will minimize the total number of possible collisions caused by the cue ball. The other player tries to find initial conditions for the cue ball to maximize the number of collisions. The value of the game is $\sqrt{n}$ (up to constants). The lower bound is based on the Erdős-Szekeres Theorem. The upper bound may be considered a generalization of the Erdős-Szekeres Theorem.

math.DS↗

arXiv2016

Ergodic Theory and Diophantine approximation for translation surfaces and linear forms

We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff ergodic theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of a classical theorem in multi-dimensional Diophantine approximation due to W. Schmidt \cite{SchmidtMetrical, SchmidtMetrical2}. The approximation argument allows us to deduce the Birkhoff genericity of almost all lattices in a certain submanifold of the space of unimodular lattices from the Birkhoff genericity of almost all lattices in the whole space and similarly for the space of affine unimodular lattices.

math.DS↗

arXiv2015

Geometry of Farey-Ford polygons

The Farey sequence is a natural exhaustion of the set of rational numbers between 0 and 1 by finite lists. Ford Circles are a natural family of mutually tangent circles associated to Farey fractions: they are an important object of study in the geometry of numbers and hyperbolic geometry. We define two sequences of polygons associated to these objects, the Euclidean and hyperbolic Farey-Ford polygons. We study the asymptotic behavior of these polygons by exploring various geometric properties such as (but not limited to) areas, length and slopes of sides, and angles between sides.

math.DS↗

arXiv2011

Lattice Point Asymptotics and Volume Growth on Teichmuller space

We apply some of the ideas of the Ph.D. Thesis of G. A. Margulis to Teichmuller space. Let x be a point in Teichmuller space, and let B_R(x) be the ball of radius R centered at x (with distances measured in the Teichmuller metric). We obtain asymptotic formulas as R tends to infinity for the volume of B_R(x), and also for for the cardinality of the intersection of B_R(x) with an orbit of the mapping class group.

math.DS↗