Density of hyperbolicity in dimension one

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 Mañé 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]: