Hedgehogs are in the limit set of a recurrent critical point.
by
Doug Childers
University of Alabama at Birmingham

Let P be a complex polynomial with Julia set J. Suppose z₀ is an irrationally neutral fixed point. Depending on the situation, let us use D to denote the maximal Siegel disk containing z₀ (in case P is linearizable about z₀, i.e. the Siegel Case), and otherwise (in the Cremer Case) set D = z₀. Recall that, according to R. Mañé, ¶D is contained in the w-limit set of some recurrent critical point (in fact, Mañé proved this result for any irrationally neutral cycle of a rational map).

Now, let us suppose that U is a simply connected, open neighborhood of [`(D)] such that P is univalent on a neighborhood of [`U]. Then, by a result of R. Pérez-Marco, there is an invariant continuum H Ì [`U], containing [`(D)], such that H ǶU ¹ Æ. Moreover, it follows that ¶H Ì J. After R. Pérez-Marco, we call such a continuum, H, a hedgehog of z₀ (with respect to P).

We show that for every hedgehog H of z₀, there exists a recurrent critical point c Î J such that ¶H Ì w(c). This result, which can be generalized to any irrationally neutral cycle of a rational map, is an extension of Mañé's result (mentioned above). In addition, let us define M to be the closure of the union of ¶H over all hedgehogs H of z₀. We note that M is a forward invariant subcontinuum of J. As a corollary to our result, if z₀ is an indifferent fixed point of a complex polynomial with exactly one critical point c, then w(c) = M.

Date received: February 28, 2005