Oleg S. Kozlovski

Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials

We prove that topologically conjugate non-renormalizable polynomials are quasi-conformally conjugate. From this we derive that each such polynomial can be approximated by a hyperbolic polynomial. As a by-product we prove that the Julia set of a non-renormalizable polynomial with only hyperbolic periodic points is locally connected, and the Branner-Hubbard conjecture. The main tools are the enhanced nest construction (developed in a previous joint paper with [Rigidity for real polynomials, Ann. of Math. (2) 165 (2007) 749-841.]) and a lemma of Kahn and Lyubich (for which we give an elementary proof in the real case).

Getting rid of the negative Schwarzian derivative condition

In this paper we will show that the assumption on the negative Schwarzian derivative is redundant in the case of C 3 unimodal maps with a non∞at critical point. The following theorem will be proved: For any C 3 unimodal map of an interval with a non∞at critical point there exists an interval around the critical value such that the flrst entry map to this interval has negative Schwarzian derivative. Another theorem proved in the paper provides useful cross-ratio estimates. Thus, all theorems proved only for unimodal maps with negative Schwarzian derivative can be easily generalized.

Stable Maps are Dense in Dimensional One

This is an exposition of our resent results contained in Kozlovski et al. (Rigidity for real polynomials, preprint, 2003; Density of hyperbolicity, preprint, 2003) and Kozlovski and van Strien (Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, preprint, 2006) where we prove the density of hyperbolicity for one dimensional real maps and non-renormalizable complex polynomials. The proofs of these results are very technical, so in this paper we try to show the main ideas on some simplified examples and also give some outlines of the proofs.