FOLIATIONS, MARKOV CHAINS AND ERGODIC THEORY

A cours of lectures by V. Kaimanovich
 

The leafwise Brownian motion on Riemannian foliations was first considered
by Lucy Garnett in 1983. Although there are foliations without holonomy
invariant measures, she pointed out that for any foliation of a compact
Riemannian manifold M there is a natural probability measure (called
harmonic) on M. Later the notion of harmonic measure found numerous
applications in works of Ghys, Brunella, Candel.

The cours is mainly based on author's work in this area with references to
results of Garnett, Zimmer, Adams, Hamenstadt, Frankel, Yue. Its aim is to
pursue an approach based on ergodic theory and general theory of Markov
processes. This approach clarifies earlier results and lead to new ones, but
requires an introductory part dedicated to the general ergodic theory
machinery.

A preliminary list of subjects to be considered:

1. Background part
Generalities on Markov chains. Examples: random walks on groups, Brownian
motion on manifolds. Harmonic functions, the Poisson boundary, the Poisson
formula. Markov chains with a finite stationary measure.

2. Main part
Harmonic measures for foliations of compact manifolds; local and global
descriptions; relations with holonomy invariant measures. Ergodicity and
mixing of the Brownian motion on foliations. Entropy of foliations and the
leafwise Liouville property; dimension of spaces of leafwise harmonic
functions; isomorphism of classes of harmonic and holonomy invariant
measures for foliations of subexponential growth. Example: foliations
associated with the geodesic flow on compact negatively curved manifolds.
Entropy and the Milnor--Wood formula. The Liouville property and
amenability of foliations.

LUGAR: Seminario do Departamento de Xeometría e Topoloxía

HORARIO:

Mércores 22: 12-13 horas
Xoves 23: 12-13 horas
Venres 24: 18-19 horas
Luns 27: 12-13 horas
Martes 28: 12-13 horas
Mércores 29: 11,30-13 horas
Xoves 30: 11,30-13 horas