Jeff Galkowski (University College London)
Weyl laws and closed geodesics on typical manifolds
Abstract:
We discuss the typical behavior of two important quantities on compact
Riemannian manifolds: the number of primitive closed geodesics of a certain
length and the error in the Weyl law. For Baire generic metrics, the qualitative
behavior of both of these quantities has been well understood since the 1970's
and 1980's. Nevertheless, their quantitative behavior for typical manifolds has
remained mysterious. In fact, only recently, Contreras proved an exponential
lower bound for the number of closed geodesics on a Baire generic manifold.
Until now, this was the only quantitative estimate on the number of geodesics
for typical metrics, and no such estimate existed for the remainder in the Weyl
law. In this talk, we give stretched exponential upper bounds on the number of
primitive closed geodesics for typical metrics. Furthermore, using recent
results on the remainder in the Weyl law, we will use our dynamical estimates to
show that logarithmic improvements in the remainder in the Weyl law hold for
typical manifolds. The notion of typicality used in this talk is a new analog of
full Lebesgue measure in infinite dimensions called predominance.
Based on joint work with Y. Canzani.