Date: Wed, 17 Nov 1999 11:36:29 +0100 To: webmaster@math.sunysb.edu From: Christian Bonatti Subject: Problem in dynamical systems. Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Length: 1512 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 Christian BONATTI Laboratoire de Topologie UMR 5584 du CNRS Universite de Bourgogne B.P. 47 870 21 078 Dijon Cedex France Fax: 03 80 39 58 99 Tel: 03 80 39 58 39 Tel: 03 80 39 58 20 (secretatiat)