Spatial and temporal central limit theorems for geodesic and horocycle flows on hyperbolic surface

07.06.2021, 14:00
Forschungsseminar Wahrscheinlichkeitstheorie

Emilio Corso (ETH Zürich)

A leitmotif in probability theory is the discovery of universal, deterministic phenomena underlying the asymptotic behaviour, in a suitably defined sense, of random structures. In the ergodic theory of discrete transformations and flows, the perspective is often reversed: while the short-term evolution of a certain system is prescribed in a deterministic fashion, the long-term behaviour of typical orbits frequently exhibits remarkable randomness features, akin to those shared by large classes of stochastic processes.

The talk surveys a specific instance of such a paradigm, where the dynamics is given by the geodesic flow on negatively curved surfaces. It will emerge that time averages of observables along geodesic orbits behave much like empirical means of independent random variables, to the extent that they satisfy an analogue of Donsker's invariance principle. Finally, we draw a comparison with the closely related horocycle flow, in which a subtler form of central limit theorem appears, namely an analogue of the Erdös-Kac theorem in probabilistic number theory.

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