Ergodic geometry in nonpositive curvature

• Applicant: Dr. Gabriele Link
• Subject Area: Mathematics
• Term: from 2007 to 2011
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 42819614

This project is devoted to a detailed study of the geodesic flow and the Weyl chamber flow in locally symmetric spaces of noncompact type, Euclidean buildings and quotients of certain CAT(0)-spaces. We want to investigate ergodic properties of these flows with respect to a class of invariant measures obtained in a standard way from generalized Patterson-Sullivan measures on the geometric boundary of the universal covering space. Such measures are particularly useful when dealing with quotients for which no finite Haar measure is available. Questions arising in this context concern ergodicity and mixing as well as the distribution of periodic orbits of the flows with respect to these measures. One important open problem for finite volume locally symmetric spaces of higher rank is to determine how the number of periodic flats of a given volume T behaves as T tends to infinity. Also related to the subject is the entropy rigidity conjecture in higher rank which, roughly, states that if a finite volume manifold carries a locally symmetric metric of nonpositive sectional curvature and higher rank, then this metric minimizes the volume entropy among all Riemannian metrics with the same volume.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1154: Global Differential Geometry