Title: Dynamical Rigidity and Geometry in Infinite Volume

Abstract: The rigidity of unipotent flows has been a central and impactful theme in homogeneous dynamics. A special instance of this phenomenon is observed in the horocycle flow on finite volume hyperbolic surfaces, which supports exactly two long-term behaviors (all orbits are either periodic or equidistributed). One striking aspect of this strong dynamical rigidity is its complete independence from the particular geometry of the ambient hyperbolic surface.

After introducing the above notions in the talk, we will examine the extent of this rigidity within surfaces of infinite volume (more specifically, geometrically infinite) and show how geometry, in fact, plays a delicate role in this setting. No specific prior knowledge will be assumed. The results are part of an ongoing series of works together with James Farre and Yair Minsky.