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Đề tài " Real polynomial diffeomorphisms with maximal entropy: Tangencies "

This paper deals with some questions about the dynamics of diffeomorphisms of R2 . A “model family” which has played a significant historical role in dynamical systems and served as a focus for a great deal of research is the family introduced by H´non, which may be written as e fa,b (x, y) = (a − by − x2 , x) b = 0. | Annals of Mathematics Real polynomial diffeomorphisms with maximal entropy Tangencies By Eric Bedford and John Smillie Annals of Mathematics 160 2004 1 26 Real polynomial diffeomorphisms with maximal entropy Tangencies By Eric Bedford and John Smillie Introduction This paper deals with some questions about the dynamics of diffeomor-phisms of R2. A model family which has played a significant historical role in dynamical systems and served as a focus for a great deal of research is the family introduced by Henon which may be written as fa b x y a - by - x2 x b 0. When b 0 fa b is a diffeomorphism. When b 0 these maps are essentially one dimensional and the study of dynamics of fa 0 reduces to the study of the dynamics of quadratic maps fa x a - x2. Like the Henon diffeomorphisms of R2 the quadratic maps of R have also played a central role in the field of dynamical systems. These two families of dynamical systems fit together naturally which raises the question of the extent to which the dynamics is similar. One difference is that our knowledge of these quadratic maps is much more thorough than our knowledge of these quadratic diffeomorphisms. Substantial progress in the theory of quadratic maps has led to a rather complete theoretical picture of their dynamics and an understanding of how the dynamics varies with the parameter. Despite significant recent progress in the theory of Henon diffeomor-phisms due to Benedicks and Carleson and many others there are still many phenomena that are not nearly so well understood in this two-dimensional setting as they are for quadratic maps. One phenomenon which illustrates the difference in the extent of our knowledge in dimensions one and two is the dependence of the complexity of the system on parameters. In one dimension the nature of this dependence is understood and the answer is summarized by the principle of monotonicity. Loosely stated this is the assertion that the complexity of fa does not Research supported in part by