The horocyclic 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. This phenomenon is now understood as part of the broader theory of unipotent rigidity on finite-volume homogeneous spaces, following Ratner.

Since the 1990s, infinite horocyclic invariant measures have been studied by Burger, Roblin, Sarig, Ledrappier, Lindenstrauss, and myself, among others. These works show that a form of rigidity persists in the infinite-measure setting for a wide class of infinite-area hyperbolic surfaces, suggesting the possibility of universal Ratner-type phenomena beyond finite volume.

In this talk, we present a construction of geometrically infinite hyperbolic surfaces in which horocycle recurrence can be prescribed with precision. As a consequence, we obtain the first examples of non-trivial, non-compact horocyclic minimal sets supporting new types of invariant measures that violate 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.