Date: Wed, 17 Nov 1999 11:36:29 +0100
From: Christian Bonatti 
Subject: Problem in dynamical systems.

As suggested in your web page I send you some problem in dynamical systems:

Topology of the hyperbolic attractors for diffeomorphism:

On surfaces, an hyperbolic attractor can be either the torus $T^2$ (Anosov
case) or a        $1-$dimensional lamination ("Plykyn attractors"). 
On 3-dimensional manifolds there are many kinds of hyperbolic attractors:
Let $A$ denote some hyperbolic attractor of some diffeomorphism $f$ on a
compact 3-manifold $M$.
- --case 1: If the unstable manifolds of the points $x\in A$ are
bidimensional: then 
          $A$ is either the torus $T^3$ (Anosov case), or a bidimensional
lamination.
- --case 2: If the unstable manifolds of the points $x\in A$ are
1-dimensional, then the attractor can be 
- --------2.a)  a 1-dimensional lamination which is transversaly
Cantor(Williams attractors)
- --------2.b)  an invariant topological 2-torus $T^2$, and the restriction
of $f$ to this torus is conjugated to an Anosov diffeomorphism (be
carefull, the torus $T^2$ can be fractal with Hausdorff dimension strickly
bigger than 2 (cf Kaplan, Mallet-Paret and Yorke). 
- --------2.??) Problem: is there some other possibility? For example, is it
possible to get an attractor such that the transversal structure of the
unstable lamination is a Sierpinsky carpet? 

Yours sincerely
Christian Bonatti

Laboratoire de Topologie 
UMR 5584 du CNRS
Universite de Bourgogne
B.P. 47 870
21 078  Dijon Cedex France