The Ergodic Theory of Discrete Groups

The particular focus of the book is on the remarkable measure supported on the limit set of a discrete group that was first developed by S. J. Patterson.

Peter J. Nicholls (Author)

9780521376747, Cambridge University Press

Paperback, published 17 August 1989

236 pages
22.8 x 15.2 x 1.4 cm, 0.352 kg

The interaction between ergodic theory and discrete groups has a long history and much work was done in this area by Hedlund, Hopf and Myrberg in the 1930s. There has been a great resurgence of interest in the field, due in large measure to the pioneering work of Dennis Sullivan. Tools have been developed and applied with outstanding success to many deep problems. The ergodic theory of discrete groups has become a substantial field of mathematical research in its own right, and it is the aim of this book to provide a rigorous introduction from first principles to some of the major aspects of the theory. The particular focus of the book is on the remarkable measure supported on the limit set of a discrete group that was first developed by S. J. Patterson for Fuchsian groups, and later extended and refined by Sullivan.

1. Preliminaries
2. The limit set
3. A measure of the limit set
4. Conformal densities
5. Hyperbolically harmonic functions
6. The sphere at infinity
7. Elementary ergodic theory
8. The geodesic flow
9. Geometrically finite groups
10. Fuchsian groups.

Subject Areas: Complex analysis, complex variables [PBKD], Groups & group theory [PBG]