### Rate of escape and SRB measures for dynamical systems with holes

Maxence Novel

Abstract.
We study dynamical systems with a open "hole" into which elements
can fall. We'll first review a result about the rate of escape of the
Riemannian measure: given some assumptions on the hole and the
hyperbolicity of our system, the rate of escape satisfies a variational
principle for pressure. Then we'll see how to extend this discrete-time
result to continuous time systems (flows), the main issue being the
flow direction, which is neither contracting or expanding. Finally, we'll
construct an SRB measure for flows and an SRB-like measure for
systems with holes and obtain a stability result: if we shrink the hole
to a point, the SRB-like measure of the open system converges to the
SRB measure of the system without hole.