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

© Fernando Alcalde.