Abstract

In the early 70's it was clear that uniform hyperbolicity concept is very strong property and the relaxing forms of hyperbolicity like partial hyperbolicity for diffeomorphisms were introduced. In this mini-course we begin with the fundamental examples (algebraic and geometric) of such dynamics and the topological restrictions that they impose on the ambient manifold by means of their action on homology. (c.f. the survey [1], and [2]).

A conjecture of Pugh and Shub on the abundance of ergodicity [3, 4] gave a prominence to the study of partially hyperbolic systems in both topological and geometrical ways. We describe the main idea to prove ergodicity which is "based" on a simple idea of accessibility by a pair of foliations. Using some well known results in the theory of 3-manifolds we conclude that on some manifolds all partially hyperbolic diffeomorphisms are ergodic [5].

Finally we describe other approaches to prove ergodicity [6] which yield to prove Pugh-Shub conjecture in a large context and show the variety of topics involved in the theory of partially hyperbolic dynamical systems.

References:

[1] F. Rodriguez Hertz and M. A. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics, Partially hyperbolic dynamics, laminations, and Teichmuller flow, Fields Inst. Communications, 51, 35-87, 2007.
[2] M. Brin and D. Burago and S. Ivanov, On partially hyperbolic diffeomorphisms on $3$-manifolds with commutative fundamental group, Advances in Dynamical Systems, Cambridge Univ. Press.
[3] C. Pugh and M. Shub, Stably ergodic dynamical systems and partial hyperbolicity, J. Complexity,13, 125--179,1997.
[4] C. Pugh and M. Shub, Stable ergodicity, With an appendix by Alexander Starkov, Bull. Amer. Math. Soc, 41, 2004.
[5] F. R. Hertz, M. A. Hertz, R. Ures, Partial hyperbolicity and ergodicity in dimension three, J. Mod. Dyn., 2, 2008, 187--208.
[6] F. R. Hertz, M. A. Hertz, A. Tahzibi, R. Ures, Criteria for ergodicity and non uniform hyperbolicity, Preprint 2009.



Information: