Kombinatorisk-Analytisk beskrivelse af Mandelbrotmængden

Description

Yoccoz has proven that the Mandelbrotset is locally connected at (harmonic measure) almost all boundary points. One of the principal tools is the puzzles now known as Yoccoz-puzzles. The idea behind Yoccoz puzzles is Markov partitions. Let $Q_c$ be a quadratic polynomial with two repelling fixed points. The essential ingredient in Yoccoz puzzles for $Q_c$ is a certain $q$-cycle, $q>1$ of external rays of the Julia set colanding at one of the fixed points $\alpha(c)$ together with all the iterated preimages of this cycle. In a joint work with Pascale Roesch we revisit Yoccoz-puzzles from the combinatorial point of view. The colanding of rays in the puzzle defines a certain equivalence relation on the corresponding set of arguments of external rays. These equivalence relations satisfy a few simple admissibility properties. We explore the space of all equivalence relations satisfying these admissibility conditions and find that essentially all such relations are realized in an essentially unique way by some quadratic polynomial.

Emneord: The Mandelbrot set, Yoccoz Puzzles, Combinatorial Encoding

Period	8 Mar 2007
Event title	Séminaire Analyse et géométrie complexes
Event type	Conference
Location	Toulouse, FranceShow on map