Absorbing Cantor sets in dynamical systems: Fibonacci maps

Abstract

In this paper we shall show that there exists a polynomial unimodal map f: [0,1] -> [0,1] which is 1) non-renormalizable(therefore for each x from a residual set, $\omega (x)$ is equal to an interval), 2) for which $\omega (c)$ is a Cantor set, and 3) for which $\omega (x)=\omega (c)$ for Lebesgue almost all x. So the topological and the metric attractor of such a map do not coincide. This gives the answer to a question posed by Milnor.

References in corpus (5)

• On the Lebesgue measure of the Julia set of a quadratic polynomial
• Ergodic theory for smooth one-dimensional dynamical systems
• Hyperbolicity is dense in the real quadratic family
• Combinatorics, geometry and attractors of quasi-quadratic maps
• Quasisymmetric conjugacies between unimodal maps