tailieunhanh - Đề tài " Toward a theory of rank one attractors "

This paper is about a class of strange attractors that have the dual property of occurring naturally and being amenable to analysis. Roughly speaking, a rank one attractor is an attractor that has some instability in one direction and strong contraction in m−1 directions, m here being the dimension of the phase space. The results of this paper can be summarized as follows. Among all maps with rank one attractors, we identify, inductively, subsets Gn, n = 1, 2, 3, · · · , | Annals of Mathematics Toward a theory of rank one attractors By Qiudong Wang andLai-Sang Young Annals of Mathematics 167 2008 349-480 Toward a theory of rank one attractors By Qiudong Wang and Lai-Sang Young Contents Introduction 1. Statement of results Part I. Preparation 2. Relevant results from one dimension 3. Tools for analyzing rank one maps Part II. Phase-space dynamics 4. Critical structure and orbits 5. Properties of orbits controlled by critical set 6. Identification of hyperbolic behavior formal inductive procedure 7. Global geometry via monotone branches 8. Completion of induction 9. Construction of SRB measures Part III. Parameter issues 10. Dependence of dynamical structures on parameter 11. Dynamics of curves of critical points 12. Derivative growth via statistics 13. Positive measure sets of good parameters Appendices Introduction This paper is about a class of strange attractors that have the dual property of occurring naturally and being amenable to analysis. Roughly speaking a rank one attractor is an attractor that has some instability in one direction and strong contraction in m 1 directions m here being the dimension of the phase space. The results of this paper can be summarized as follows. Among all maps with rank one attractors we identify inductively subsets Gn n 1 2 3 Both authors are partially supported by grants from the NSF. 350 QIUDONG WANG AND LAI-SANG YOUNG consisting of maps that are well-behaved up to the nth iterate. The maps in Q nn oGn are then shown to be nonuniformly hyperbolic in a controlled way and to admit natural invariant measures called SRB measures. This is the content of Part II of this paper. The purpose of Part III is to establish existence and abundance. We show that for large classes of 1-parameter families Ta Ta e Q for positive measure sets of a. Leaving precise formulations to Section 1 we first put our results into perspective. A. In relation to hyperbolic theory. Axiom A theory together with its extension to