Genadi Levin

LOCAL CONNECTIVITY OF THE JULIA SET OF REAL POLYNOMIALS

In particular, the Julia set of z + c1 is locally connected if c1 ∈ [−2, 1/4] and totally disconnected if c1 ∈ R \ [−2, 1/4] (note that [−2, 1/4] is equal to the set of parameters c1 ∈ R for which the critical point c = 0 does not escape to infinity). This answers a question posed by Milnor, see [Mil1]. We should emphasize that if the ω-limit set ω(c) of the critical point c = 0 is not minimal then it very easy to see that the Julia set is locally connected, see for example Section 10. Yoccoz [Y] already had shown that each quadratic polynomial which is only finitely often renormalizable (with non-escaping critical point and no neutral periodic point) has a locally connected Julia set. Moreover, Douady and Hubbard [DH1] already had shown before that each polynomial of the form z 7→ z + c1 with an attracting or neutral parabolic cycle has a locally connected Julia set. As will become clear, the difficult case is the infinitely renormalizable case. In fact, using the reduction method developed in Section 3 of this paper, it turns out that in the non-renormalizable case the Main Theorem follows from some results in [Ly3] and [Ly5], see the final section of this paper.

An inequality for laminations, Julia sets and `growing trees'

For a closed lamination on the unit circle invariant under z\mapsto z^d we prove an inequality relating the number of points in the ‘gaps’ with infinite pairwise disjoint orbits to the degree; in particular, this gives estimates on the cardinality of any such ‘gap’ as well as on the number of distinct grand orbits of such ‘gaps’. As a tool, we introduce and study a dynamically defined growing tree in the quotient space. We also use our techniques to obtain for laminations an analog of Sullivans no wandering domain theorem. Then we apply these results to Julia sets of polynomials.

External rays to periodic points

We prove that for every polynomial-like holomorphic mapP, ifaεK (filled-in Julia set) and the componentKaofK containinga is either a point ora is accessible along a continuous curve from the complement ofK andKais eventually periodic, thena is accessible along an external ray. Ifa is a repelling or parabolic periodic point, then the set of arguments of the external rays converging toa is a nonempty closed “rotation set”, finite (ifKais not a one point) or Cantor minimal containing a pair of arguments of external rays of a critical point in ℂ. In the Appendix we discuss constructions via cutting and glueing, fromP to its external map with a “hedgehog”, and backward.

When do two rational functions have the same Julia set

It is proved that non-exceptional rational functions f and g on the Riemann sphere have the same measure of maximal entropy iff there exist iterates F of f and G of g and natural numbers M, N such that (*) (G-1oG)oGM=(F-1oF)o FN. If one assumes only that f, g have the same Julia set and no singular or parabolic domains of normality for the iterates, one also proves (*).

On explicit connections between dynamical and parameter spaces

We study the velocity of motion of periodic orbits of polynomial and polynomial-like families. Applications to the Mandelbrot sets are given.

Ruelle operators with rational weights for Julia sets

AbstractLetR be a rational function with nonempty set of normality that consists of basins of attraction only and let

Multipliers of periodic orbits in spaces of rational maps

\left( {L_Q g} \right)\left( z \right) = \sum\limits_{R\left( w \right) = z} {Q\left( w \right)g\left( w \right)}

On the complement of the Mandelbrot set

be a Ruelle operator with a rational weightQ which acts in a space of locally analytic functions on the Julia set ofR. We obtain explicit expressions for equations for the eigenvalues and study the structure of eigenfunctions of LQ and its adjoint operator. Some applications to operators important for measurable dynamics of rational functions will be considered.