Wild Cantor attractors exist


In this paper we shall show that there exists a polynomial unimodal map $f\colon [0,1] \to [0,1]$ with so-called Fibonacci dynamics

$\bullet$ which is non-renormalizable and in particular, for each $x$ from a residual set $\omega(x)$ is equal to an interval (here $\omega(x)$ is defined to be the set of accumulation points of the sequence $x$, $f(x)$, $f^2(x),\dots)$;

$\bullet$ for which the closure of the forward orbit of the critical point $c$, i.e., $\omega(c)$, is a Cantor set and

$\bullet$ or which $\omega(x) = \omega(c)$ for Lebesgue-almost all $x$.

So the topological attractor and the metric attractor of such a map do not coincide. This gives the answer to a question posted by Milnor [Mil] in dimension one.




Hendrik Bruin

Gerhard Keller

Tomasz Nowicki

Sebastian van Strien