TAILIEUCHUNG - Đề tài " Density of hyperbolicity in dimension one "

Annals of Mathematics In this paper we will solve one of the central problems in dynamical systems: Theorem 1 (Density of hyperbolicity for real polynomials). Any real polynomial can be approximated by hyperbolic real polynomials of the same degree. Here we say that a real polynomial is hyperbolic or Axiom A, if the real line is the union of a repelling hyperbolic set, the basin of hyperbolic attracting periodic points and the basin of infinity. We call a C 1 endomorphism of the compact interval (or the circle) hyperbolic if it has finitely many hyperbolic attracting periodic points and the. | Annals of Mathematics Density of hyperbolicity in dimension one By O. Kozlovski W. Shen and S. van Strien Annals of Mathematics 166 2007 145 182 Density of hyperbolicity in dimension one By O. Kozlovski W. Shen and S. VAN Strien 1. Introduction In this paper we will solve one of the central problems in dynamical systems Theorem 1 Density of hyperbolicity for real polynomials . Any real polynomial can be approximated by hyperbolic real polynomials of the same degree. Here we say that a real polynomial is hyperbolic or Axiom A if the real line is the union of a repelling hyperbolic set the basin of hyperbolic attracting periodic points and the basin of infinity. We call a C1 endomorphism of the compact interval or the circle hyperbolic if it has finitely many hyperbolic attracting periodic points and the complement of the basin of attraction of these points is a hyperbolic set. By a theorem of Mane for C2 maps this is equivalent to the following conditions all periodic points are hyperbolic and all critical points converge to periodic attractors. Note that the space of hyperbolic maps is an open subset in the space of real polynomials of fixed degree and that every hyperbolic map satisfying the mild no-cycle condition which states that orbits of critical points are disjoint is structurally stable see dMvS93 . Theorem 1 solves the 2nd part of Smale s eleventh problem for the 21st century Sma00 Theorem 2 Density of hyperbolicity in the Ck topology . Hyperbolic . Axiom A maps are dense in the space of Ck maps of the compact interval or the circle k 1 2 . X IV. This theorem follows from the previous one. Indeed one can approximate any smooth or analytic map on the interval by polynomial maps and therefore by Theorem 1 by hyperbolic polynomials. Similarly one can approximate any map of the circle by trigonometric polynomials. If a circle map does not have periodic points it is semi-conjugate to the rotation and it can be approximated by an Axiom A map this is a .

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