Bedford T., Keane M., Series C. (eds.) Ergodic theory, symbolic dunamics and hyperbolic spaces

Oxford: Oxford University Press, 1991. — 385 p.This volume originated as a series of lectures given at the workshop Hyperbolic Geometry and Ergodic Theory' held at the International Centre for Theoretical Physics in Trieste, 17-28 April, 1989. Hyperbolic geometry and surfaces of negative curvature were crucial to the early development of ergodic theory, so the two subjects have a long history of interaction, in fact the geodesic flow on a surface of constant negative curvature was probably the first seriously non-trivial example of an ergodic flow. The relationship of the two subjects is particularly close in the area of the dynamics and coding of geodesic flows. This topic is our main underlying theme. For example, the dynamical problem of counting periodic orbits for the geodesic flow is equivalent to the group theoretic problem of counting conjugacy classes in the fundamental group of the surface in question.
By using the theory of Markov partitions to code the geodesic flow, both problems are reduced to a combinatorial problem about the number of periodic points in a certain shift space (a subshift of finite type), which is solved by methods originally derived from statistical mechanics. In many cases the limit set of the fundamental group can also be coded as a subshift of finite type. The current very interesting work on the recursive structure of hyperbolic groups is closely linked with this idea. Patterson's construction of a measure on the limit set (which allows generalisations to higher dimensions of many of the early ergodic theoretic results) can also be put into this context.
The volume contains three chapters at introductory level followed by eight more advanced ones based on the earlier material. At the meeting there were also a number of specialist lectures, of which we have included only the one (Chapter 10) pertaining directly to the topics of the main courses. Each chapter is an expanded version of the lectures given by its author. While there has been some attempt to co-ordinate and cross-reference, the diverse nature of the chapters reflects the styles of the individual lecturers, and there is no claim to a comprehensive treatment.