# The Yoccoz Combinatorial Analytic Invariant

Carsten Lunde Petersen, Pascale Roesch

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### Resumé

I denne artilel udvikler vi en kombinatorisk-analytisk adressering af Mandelbrotmængden M. Denne adressering findes implicit i Yoccoz bevis for localsammenhæng af M i enhver Yoccoz parameter, i.e. parametre som er højest endeligt mange gange renormaliserbare og for hvilke enhver periodisk bane er frastødende. Ved brug af denne adressering definerer vi en eksplicit combinatorisk-analytisk model, der er tilstrækkelig abstrakt til at den kan være en tro model af både M og den parabolske Mandelbrotmængde M1. Som en umiddelbar anvendelse finder vi et nyt bevis for at vener i M er stier og at mere generelt at ethvert par af Yoccoz parametre er forbundet i M af en entydigt bestemt Douady-Hubbard regular sti.

Originalsprog Engelsk Fields Institute Communications 53 145-176 32 1069-5265 Udgivet - 2008

### Bibliografisk note

Udgivet i:
Holomorphic dynamics and renormalization : a volume in honour of John Milnor's 75th birthday / Mikhail Lyubich, Michael Yampolsky, editors. - Providence, R.I. : American Mathematical Society ; Toronto, Ont. : Fields Institute for Research in Mathematical Sciences, c2008
ISBN-10: 0-8218-4275-7
ISBN-13: 978-0-8218-4275-1

### Emneord

• Mandelbrotmængden
• Yoccoz Puzzles
• Kombinatorisk Kodning

### Citer dette

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title = "The Yoccoz Combinatorial Analytic Invariant",
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.",
keywords = "Mandelbrotm{\ae}ngden, Yoccoz Puzzles, Kombinatorisk Kodning, The Mandelbrot set, Yoccoz Puzzles, Combinatorial Encoding",
author = "Petersen, {Carsten Lunde} and Pascale Roesch",
note = "Udgivet i: Holomorphic dynamics and renormalization : a volume in honour of John Milnor's 75th birthday / Mikhail Lyubich, Michael Yampolsky, editors. - Providence, R.I. : American Mathematical Society ; Toronto, Ont. : Fields Institute for Research in Mathematical Sciences, c2008 ISBN-10: 0-8218-4275-7 ISBN-13: 978-0-8218-4275-1",
year = "2008",
language = "English",
volume = "53",
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journal = "Fields Institute Communications",
issn = "1069-5265",
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The Yoccoz Combinatorial Analytic Invariant. / Petersen, Carsten Lunde; Roesch, Pascale.

I: Fields Institute Communications, Bind 53, 2008, s. 145-176.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - The Yoccoz Combinatorial Analytic Invariant

AU - Petersen, Carsten Lunde

AU - Roesch, Pascale

N1 - Udgivet i: Holomorphic dynamics and renormalization : a volume in honour of John Milnor's 75th birthday / Mikhail Lyubich, Michael Yampolsky, editors. - Providence, R.I. : American Mathematical Society ; Toronto, Ont. : Fields Institute for Research in Mathematical Sciences, c2008 ISBN-10: 0-8218-4275-7 ISBN-13: 978-0-8218-4275-1

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

KW - Mandelbrotmængden

KW - Yoccoz Puzzles

KW - Kombinatorisk Kodning

KW - The Mandelbrot set

KW - Yoccoz Puzzles

KW - Combinatorial Encoding

M3 - Journal article

VL - 53

SP - 145

EP - 176

JO - Fields Institute Communications

JF - Fields Institute Communications

SN - 1069-5265

ER -