04/12: Semyon Dyatlov
Ruelle zeta at zero for nearly hyperbolic 3-manifolds
Abstract: For a compact negatively curved Riemannian manifold
, the Ruelle zeta function
of its geodesic flow is defined for
as a convergent product over the periods
of primitive closed geodesics
and extends meromorphically to the entire complex plane. If
is hyperbolic (i.e. has sectional curvature
), then the order of vanishing
of
at
can be expressed in terms of the Betti numbers
. In particular, Fried proved in 1986 that when
is a hyperbolic 3-manifold,
I will present a recent result joint with Mihajlo Cekić, Benjamin Küster, and Gabriel Paternain: when
and
is a generic perturbation of the hyperbolic metric, the order of vanishing of the Ruelle zeta function jumps, more precisely
This is in contrast with dimension 2 where
for all negatively curved metrics. The proof uses the microlocal approach of expressing
as an alternating sum of the dimensions of the spaces of generalized resonant Pollicott–Ruelle currents and obtains a detailed picture of these spaces both in the hyperbolic case and for its perturbations.