March 20, 1997. M.I.Gordin. Homoclinic structtures in hyperbolic dynamics

Anosov diffeomorphisms (and, more generally, Smale diffeomorphisms) and topological Markov chains are examples of HDS' for the case of Z action ('discret time'). HDS' with continous time (i.e. with R action) include geodesic flows on riemannian manifolds of negative sectional curvature.

The most powerful tool to study HDS' (especially their probabilistic properties) is the symbolic dynamics (more precisely, the theory of Markov partitions). Though the main part of known results concerning HDS' is based on this technique, it has also some drawbacks: noncanonicity of partitions and objects derived from them, difficulties with extension of this technique to wider classes of dynamical systems

There exists an alternative approach to hyperbolic dynamics based on homoclinic equivalence relations and homoclinic grouppoids, both of them are associated with HDS' in a canonic way. Though this approach is not yet developed enough to eliminate Markov partitions, some interesting applications of homoclinic structures are known.

In the talk the following subjects are supposed to be concerned:

• homoclinic structures as topological invariants of HDS';
• the concept of Gibbsian measure within homoclinic setting;
• application of homoclinic structures to the spectral theory;
• 'homoclinic laplacian' for hyperbolic toral automorphisms
• and its application to the central limit theorem;
• interrelation with operator algebras.

List of talks at previous sessions of the seminar.