Density of hyperbolicity in dimension one. (English) Zbl 1138.37013

Kozlovski, O.; Shen, W.; van Strien, S.

Ann. Math. (2) 166, No. 1, 145-182 (2007).

The authors investigate one of the central problems in dynamical systems – density of hyperbolicity in \(C^k\) topology – which is the second part of Smale’s eleventh problem for the 21st century. The main result of the paper is: Any real polynomial can be approximated by hyperbolic real polynomials of the same degree.
Here the authors 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. The working tool in proving the main theorem is the existence of the so called box mappings and their properties.

Reviewer: Alois Klíč (Praha)

MSC:

37D20	Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37E05	Dynamical systems involving maps of the interval
37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F20	Combinatorics and topology in relation with holomorphic dynamical systems

Keywords:

hyperbolicity; Axiom A maps; critical points; combinatorial equivalence; quasi-conformal conjugacy; Julia set