Hölder implies Collet-Eckmann. (English) Zbl 0939.37026

Przytycki, Feliks

Flexor, Marguerite (ed.) et al., Géométrie complexe et systèmes dynamiques. Colloque en l’honneur d’Adrien Douady à l’occasion du soixantième anniversaire, Orsay, France, du 3 au 8 juillet 1995. Paris: Astérisque, Astérisque. 261, 385-403 (2000).

Summary: The author proves that for every polynomial \(f\) if its basin of attraction to \(\infty\) is Hölder and the Julia set contains only one critical point \(c\) then \(f\) is Collet-Eckmann, namely there exists \(\lambda>1\), \(C>0\) such that, for every \(n\geq 0\), \(|(f^n)'(f(c))|\geq C\lambda^n\). He introduces also topological Collet-Eckmann rational maps and repellers.
For the entire collection see [Zbl 0932.00046].

MSC:

37F50	Small divisors, rotation domains and linearization in holomorphic dynamics
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37L45	Hyperbolicity; Lyapunov functions for infinite-dimensional dissipative dynamical systems

Keywords:

Collet-Eckmann holomorphic maps; repellers; Hölder basin of attraction; non-uniformly hyperbolic; telescope; rational maps; iteration; Julia set