496 research outputs found
Local connectivity of the Mandelbrot set at some satellite parameters of bounded type
We explore geometric properties of the Mandelbrot set M, and the
corresponding Julia sets J_c, near the main cardioid. Namely, we establish
that: a) M is locally connected at certain infinitely renormalizable parameters
c of bounded satellite type, providing first examples of this kind; b) The
Julia sets J_c are also locally connected and have positive area; c) M is
self-similar near Siegel parameters of constant type. We approach these
problems by analyzing the unstable manifold of the pacman renormalization
operator constructed in [DLS] as a global transcendental family
Algorithmic aspects of branched coverings
This is the announcement, and the long summary, of a series of articles on
the algorithmic study of Thurston maps. We describe branched coverings of the
sphere in terms of group-theoretical objects called bisets, and develop a
theory of decompositions of bisets.
We introduce a canonical "Levy" decomposition of an arbitrary Thurston map
into homeomorphisms, metrically-expanding maps and maps doubly covered by torus
endomorphisms. The homeomorphisms decompose themselves into finite-order and
pseudo-Anosov maps, and the expanding maps decompose themselves into rational
maps.
As an outcome, we prove that it is decidable when two Thurston maps are
equivalent. We also show that the decompositions above are computable, both in
theory and in practice.Comment: 60-page announcement of 5-part text, to apper in Ann. Fac. Sci.
Toulouse. Minor typos corrected, and major rewrite of section 7.8, which was
studying a different map than claime
MLC at Feigenbaum points
We prove {\em a priori} bounds for Feigenbaum quadratic polynomials, i.e.,
infinitely renormalizable polynomials of bounded type. It
implies local connectivity of the corresponding Julia sets and MLC
(local connectivity of the Mandelbrot set \Mandel) at the corresponding
parameters . It also yields the scaling Universality, dynamical and
parameter, for the corresponding combinatorics. The MLC Conjecture was open for
the most classical period-doubling Feigenbaum parameter as well as for the
complex tripling renormalizations. Universality for the latter was conjectured
by Goldberg-Khanin-Sinai in the early 1980s
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