A famous result of H. Furstenberg says that for a compact surface S of curvature -1 the horocicles are equi-distributed inside its unit tangent bundle. Equivalently, Furstenberg theorem says that the horocyclic flow is uniquely ergodic. This has been extended to more general situations, as surfaces of variable negative curvature (B. Marcus), and horocyclic foliations of hyperbolic flows (B. Marcus and R. Bowen), to name a few.

One fruitful way to state the above results is saying that horocyclic foliations of hyperbolic systems admit one and only one invariant transverse measure; this permits to introduce the techniques of thermodynamic formalism to attack the problem, by linking the mentioned transverse measure with a special invariant probability for the hyperbolic defining dynamical system, namely the so called entropy maximizing measure.

From the geometrical point of view however, it is natural to consider not only invariant measures but also quasi-invariant ones (transverse measures which remain equivalent under the action of the holonomy pseudo-group): in this situation, is much less clear what is the relation of these with the hyperbolic dynamics.

The aim of this mini-course is to address the previous problem, elucidating the relation of quasi-invariant measures for horocyclic foliations with equilibrium states for the subjacent hyperbolic dynamics. The material originates from some recent results of a joint project with Federico Rodriguez-Hertz (PSU).