Related papers: Geometry of quadratic polynomials: moduli, rigidit…
We prove a priori bounds for Feigenbaum quadratic polynomials, i.e., infinitely renormalizable polynomials $f_c: z\mapsto z^2+c$ of bounded type. It implies local connectivity of the corresponding Julia sets $J(f_c)$ and MLC (local…
We prove combinatorial rigidity of infinitely renormalizable unicritical polynomials, P_c :z \mapsto z^d+c, with complex c, under the a priori bounds and a certain "combinatorial condition". This implies the local connectivity of the…
The renormalization of a quadratic-like map is studied. The three-dimensional Yoccoz puzzle for an infinitely renormalizable quadratic-like map is discussed. For an unbranched quadratic-like map having the {\sl a priori} complex bounds, the…
In this Note, we present recent developments in the Renormalization Theory of quadratic polynomials and discuss their applications, with an emphasis on the MLC conjecture, the problem of local connectivity of the Mandelbrot set, and on its…
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…
For an infinitely renormalizable quadratic map $f_c: z\mapsto z^2+c$ with the sequence of renormalization periods ${k_m}$ and rotation numbers ${t_m=p_m/q_m}, we prove that if $\limsup k_m^{-1}\log |p_m|>0$, then the Mandelbrot set is…
We prove that any unicritical polynomial $f_c:z\mapsto z^d+c$ which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. It implies that the connectedness locus (the ``Multibrot set'') is…
We prove the a priori bounds for infinitely renormalizable quadratic polynomials for which we can find an infinite sequence of primitive renormalizations such that the ratios of the periods of successive renormalizations is bounded. This…
In this paper we prove {\it a priori bounds} for infinitely renormalizable quadratic polynomials satisfying a ``molecule condition''. Roughly speaking, this condition ensures that the renormalization combinatorics stay away from the…
This is a continuation of the series of notes on the dynamics of quadratic polynomials. We show the following Rigidity Theorem: Any combinatorial class contains at most one quadratic polynomial satisfying the secondary limbs condition with…
We extend Sullivan's complex a priori bounds to real quadratic polynomials with essentially bounded combinatorics. Combined with the previous results of the first author, this yields complex bounds for all real quadratics. Local…
In this paper we describe the well studied process of renormalization of quadratic polynomials from the point of view of their natural extensions. In particular, we describe the topology of the inverse limit of infinitely renormalizable…
The famous MLC Conjecture states that the Mandelbrot set is locally connected, and it is considered by many to be the central conjecture in one-dimensional complex dynamics. Among others, it implies density of hyperbolicity in the quadratic…
For the study of the 2-dimensional space of cubic polynomials, J. Milnor considers the complex 1-dimensional slice S_n of the cubic polynomials which have a super-attracting orbit of period n. He gives in [M4] a detailed conjectural picture…
We study the bifurcation loci of quadratic (and unicritical) polynomials and exponential maps. We outline a proof that the exponential bifurcation locus is connected; this is an analog to Douady and Hubbard's celebrated theorem that (the…
A model for the Mandelbrot set is due to Thurston and is stated in the language of geodesic laminations. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating…
About 35 years ago Dennis Sullivan proved a precompactness property ("complex bounds") for infinitely renormalizable real quadratic polynomials with bounded combinatorics. We present a simple "soft" proof of this remarkable result.
A decoration of the Mandelbrot set $M$ is a part of $M$ cut off by two external rays landing at some tip of a satellite copy of $M$ attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials…
We establish a version of the Pommerenke-Levin-Yoccoz inequality for the modulus of a polynomial-like restriction of a global polynomial and give two applications. First it is shown that if the modulus of a polynomial-like restriction of an…
The following notes provide an introduction to recent work of Branner, Hubbard and Yoccoz on the geometry of polynomial Julia sets. They are an expanded version of lectures given in Stony Brook in Spring 1992. I am indebted to help from the…