The Yoccoz Combinatorial Analytic Invariant

Carsten Lunde Petersen, Pascale Roesch

Research output: Contribution to journalJournal articlepeer-review

Abstract

In this paper we develop a combinatorial analytic encoding of the Mandelbrot set M. The encoding is implicit in Yoccoz' proof of local connectivity of M at any Yoccoz parameter, i.e. any at most finitely renormalizable parameter for which all periodic orbits are repelling. Using this encoding we define an explicit combinatorial analytic modelspace, which is sufficiently abstract that it can serve as a go-between for proving that other sets such as the parabolic Mandelbrot set M1 has the same combinatorial structure as M. As an immediate application we use here the combinatorial-analytic model to reprove that the dyadic veins of M are arcs and that more generally any two Yoccoz parameters are joined by a unique ruled (in the sense of Douady-Hubbard) arc in M.

Original languageEnglish
Book seriesFields Institute Communications
Volume53
Pages (from-to)145-176
Number of pages32
ISSN1069-5265
Publication statusPublished - 2008

Keywords

  • The Mandelbrot set
  • Yoccoz Puzzles
  • Combinatorial Encoding

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