NON-WEAKLY TAME SURFACES: ERGODICITY AND CONSERVATIVITY OF THE HOROCYCLE FLOW

For some classes of hyperbolic surfaces, all locally finite ergodic measures  invariant under the horocycle flow are described. See the works of Omri Sarig and those of Lindenstrauss-Landesberg in that sense. This talk focuses on the study of another class of hyperbolic surfaces. Precisely, we give an analytical construction of a family of surfaces of infinite type whose corresponding horocycle flow is conservative but not ergodic with respect to the Liouville measure.