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Large Hyperbolic Circles by Corso, Emilio; Ravotti, Davide is a scholarly article available to read on EtoBox.

What is Large Hyperbolic Circles about?

We consider circles of common centre and increasing radius on a compact hyperbolic surface and, more generally, on its unit tangent bundle. We establish a precise asymptotics for their rate of equidistribution. Our result holds for translates of any circle arc by arbitrary elements of $\text{SL}_2(\mathbb{R})$. Our proof relies on a spectral method pioneered by Ratner and subsequently developed by Burger in the study of geodesic and horocycle flows. We further derive statistical limit theorems, with compactly supported limiting distribution, for appropriately rescaled circle averages of sufficient regular observables. Finally, we discuss applications to the classical circle problem in the hyperbolic plane, following the approach of Duke-Rudnick-Sarnak and Eskin-McMullen.

Author

Corso, Emilio; Ravotti, Davide

Published

2022

Language

EN