Title: Counterexamples in Horocyclic Dynamics
Speaker: Or Landesberg (HUJI)
Abstract:
The horocycle flow on a finite-area hyperbolic surface is a prototypical example of dynamical rigidity: all orbit closures and ergodic measures arise from an algebraic origin. Since the 1990s, results of Burger, Roblin, Sarig, Ledrappier, and of Lindenstrauss and myself have shown that this rigidity extends to the infinite-measure setting across a broad class of infinite-area surfaces, suggesting the possibility of a Ratner-type rigidity theory in that context.
In this talk, we present a construction of geometrically infinite hyperbolic surfaces that contain horocycles with prescribed recurrence. In particular, we obtain the first examples of non-trivial, non-compact horocyclic minimal sets carrying new types of invariant measures which violate previously exhibited infinite-measure rigidity. Further examples include highly irregular horocyclic orbit closures having fractional Hausdorff dimensions.
Based on joint work with Francoise Dal'bo, James Farre, and Yair Minsky.