Dynamical Systems Seminar

Rigidity for time-changes of unipotent flows

by Mauro Artigiani (Universidad del Rosario)

SNS - Centro De Giorgi

SNS - Centro De Giorgi

Parabolic flows form an intermediate category between elliptic and hyperbolic flows. They exhibit some characteristics associated with non-chaotic systems, and some associated with highly chaotic ones. A fundamental example is the horocycle flow on hyperbolic surfaces and, more generally, homogeneous flows generated by multiplication by a unipotent element on a Lie group. It is much more difficult to produce examples of non-homogeneous parabolic flows, since perturbations usually lead to hyperbolic flows.

The simplest perturbation, still in the parabolic realm, is given by time-changes. These have been investigated in detail in the case of the horocycle flow and for nilflows. In this talk, we will give a detailed introduction to the classical theory of horocycles and their time-changes, before presenting our result, joint with Livio Flaminio and Davide Ravotti, on rigidity of time-changes of unipotent flows on finite volume quotients of simple Lie groups, which generalizes Ratner's classical work on the horocycle flow.