Jayadev S. Athreya

Lattice point asymptotics and volume growth on Teichmüller space

We apply some of the ideas of the Ph.D. Thesis of G. A. Margulis (Mar70) to Teichmuller space. Let X be a point in Teichmuller space, and let BR(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 BR(X), and also for the cardinality of the intersection of BR(X) with an orbit of the mapping class group.

DEVIATION OF ERGODIC AVERAGES FOR RATIONAL POLYGONAL BILLIARDS

We prove a polynomial upper bound on the deviation of er- godic averages for almost all directional flows on every tran slation sur- face, in particular, for the generic directional flow of bill iards in any Euclidean polygon with rational angles.

Cusp excursions on parameter spaces

We prove several results for dynamics of SL(d,R)-actions on non-compact parameter spaces by studying associated discrete sets in Euclidean spaces. This allows us to give elementary proofs of logarithm laws for horocycle flows on hyperbolic surfaces and moduli spaces of flat surfaces. We also give applications to quantitative equidistribution and Diophantine approximation.

On the asymptotics of discrete order statistics

Let X1,X2,...,Xn be a sequence of independent, identically distributed positive integer random variables. We study the asymptotics of the likelihood that the sample maximum is achieved k times and in its spacing relative to the second highest value. Earlier and other results are discussed in context. Also, some investigation is made when the sample is Markovian. Different results emerge in this case.

Spiraling of approximations and spherical averages of Siegel transforms

J.S.A. partially supported by NSF grant DMS 1069153, and NSF grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). A.G. partially supported by the Royal Society. J.T. acknowledges the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 291147.

Ergodic properties of compositions of interval exchange maps and rotations

We study the ergodic properties of compositions of interval exchange transformations (IETs) and rotations. We show that for any IET T, there is a full measure set of α ∈ [0, 1) so that T Rα is uniquely ergodic, where Rα is rotation by α.