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New insights into the combinatorial structure of the the Mandelbrot set are given by 'Correspondence' and 'Translation' Principles both conjectured and partially proved by E. Lau and D. Schleicher. We provide complete proofs of these principles and discuss results related to them.
Note: The 'Translation' and 'Correspondence' Principles given earlier turned out to be false in the general case. In April 1999, an errata was added to discuss which parts of the two statements are incorrect and which parts remain true.
We give a new proof that all external rays of the Mandelbrot set at rational angles land, and of the relation between the external angle of such a ray and the dynamics at the landing point. Our proof is different from the original one, given by Douady and Hubbard and refined by Lavaurs, in several ways: it replaces analytic arguments by combinatorial ones; it does not use complex analytic dependence of the polynomials with respect to parameters and can thus be made to apply for non-complex analytic parameter spaces; this proof is also technically simpler. Finally, we derive several corollaries about hyperbolic components of the Mandelbrot set.
Along the way, we introduce partitions of dynamical and parameter planes which are of independent interest, and we interpret the Mandelbrot set as a symbolic parameter space of kneading sequences and internal addresses.
It will be shown that the smooth conjugacy class of an $S-$unimodal map which does not have a periodic attractor neither a Cantor attractor is determined by the multipliers of the periodic orbits. This generalizes a result by M.Shub and D.Sullivan for smooth expanding maps of the circle.
In $\textit {HO1}$, it was shown that there is a topology on $\mathbb{C}^2 \sqcup S^3$ homeomorphic to a 4-ball such that the Hénon mapping extends continuously. That paper used a delicate analysis of some asymptotic expansions, for instance, to understand the structure of forward images of lines near infinity. The computations were quite difficult, and it is not clear how to generalize them to other rational maps.
In this paper we will present an alternative approach, involving blow-ups rather than asymptotics. We apply it here only to Hénon mappings and their compositions, but the method should work quite generally, and help to understand the dynamics of rational maps $f:\mathbb{P}^2\sim\kern-2pt\sim\kern-2pt>\mathbb{P}^2$ with points of indeterminacy. The application to compositions of Hénon maps proves a result suggested by Milnor, involving embeddings of solenoids in $S^3$ which are topologically different from those obtained from Hénon mappings.
This is a preliminary investigation of the geometry and dynamics of rational maps with only two critical points. (originally titled "On Bicritical Rational Maps"; revised April 1999)
Let ${\mathcal H}$ be a hyperbolic component of quadratic rational maps possessing two distinct attracting cycles. We show that ${\mathcal H}$ has compact closure in moduli space if and only if neither attractor is a fixed point.
We prove uniform hyperbolicity of the renormalization operator for all possible real combinatorial types. We derive from it that the set of infinitely renormalizable parameter values in the real quadratic family $P_c: x \mapsto x^2+c$ has zero measure. This yields the statement in the title (where "regular" means to have an attracting cycle and "stochastic" means to have an absolutely continuous invariant measure). An application to the MLC problem is given.
In this paper we give a combinatorial description of the renormlization limits of infinitely renormalizable unimodal maps with $\textit {essentially bounded}$ combinatorics admitting quadratic-like complex extensions. As an application we construct a natural analogue of the period-doubling fixed point. Dynamical hairiness is also proven for maps in this class. These results are proven by analyzing $\textit {parabolic towers}$: sequences of maps related either by renormalization or by $\textit {parabolic renormalization}$.
Thurston's ending lamination conjecture proposes that a finitely-generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free two-generator Kleinian groups with parabolic commutator, which should be thought of as representations of the fundamental group of a punctured torus.
As a consequence we verify the conjectural topological description of the deformation space of punctured-torus groups (including Bers' conjecture that the quasi-Fuchsian groups are dense in this space) and prove a rigidity theorem: two punctured-torus groups are quasi-conformally conjugate if and only if they are topologically conjugate.
By considering the way an n-tuple of points in the 2-disk are linked together under iteration of an orientation preserving diffeomorphism, we construct a dynamical cocycle with values in the Artin braid group. We study the asymptotic properties of this cocycle and derive a series of topological invariants for the diffeomorphism which enjoy rich properties.