On quasi-conformal (in-) compatibility of satellite copies of the Mandelbrot set: I

Bidragets oversatte titel: Om quasikonform (in) kompatibilitet af satelit kopier af Mandelbrotmængden I:

Luna Lomonaco, Carsten Lunde Petersen

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Resumé

Douady and Hubbard introducerede begrebet polynomiums-lignende afbildninger i artiklen (Ann Sci Ec Norm Suppl 4 18(2):287–343, 1985) . De brugte sådanne afbildninger til at identificere homeomorfe kopier M′ af Mandelbrotmængden som egentlige delmængder af Mandelbrotmængden M selv. Disse kopier kan enten være såkaldt primitive (med en rodcusp) eller satellit kopie (uden en rodcusp). Douady og Hubbard formodede at de primitive kopier er quasikonformt homeomorfe med M, og at satellit kopierne er quasikonformt homeomorfe med M uden for enhver omegn om rodent. Disse formodninger er nu sætning bevist af Lyubich (Ann Math 149:319–420, 1999). Satelit kopierne Mp/q er tydeligt vis ikke q-c homeomorfe med M. Spørgsmålet har været om de er indbyrdes q-c homeomorfe? Eller endda q-c homeomorfe med halvdelen af den logistike Mandelbrot mængde? I denne artikel beviser vi at i almindelighed er den inducede Douady–Hubbard homeomorfi ikke
restriction of a q-c homeomorfi: For to vilkårlig satellit kopier M′ og M″ er den induced Douady–Hubbard homeomorfi ikke q-c, fikspunkts egenværdierne hørende til rod parametrene λ′=e2πip′/q′ og λ″=e2πip″/q″ q'≠q″.
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OriginalsprogEngelsk
TidsskriftInventiones Mathematicae
Vol/bind210
Udgave nummer2
Sider (fra-til)615-644
Antal sider30
ISSN0020-9910
DOI
StatusUdgivet - 2017

Emneord

  • Mandelbrotmængden

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abstract = "Douady and Hubbard (Ann Sci Ec Norm Suppl 4 18(2):287–343,1985) introduced the notion of polynomial-like maps. They used it to identifyhomeomorphic copies M of the Mandelbrot set inside the Mandelbrotset M. These copies can be primitive (with a root cusp) or satellite (withouta root cusp). They conjectured that the primitive copies are quasiconformallyhomeomorphic to M, and that the satellite ones are quasiconformallyhomeomorphic to M outside any small neighbourhood of the root. These conjectures are now Theorems due to Lyubich (Ann Math 149:319–420, 1999).The satellite copies Mp/q are clearly not q-c homeomorphic to M. But arethey mutually q-c homeomorphic? Or even q-c homeomorphic to half of thelogistic Mandelbrot set? In this paper we prove that, in general, the inducedDouady–Hubbard homeomorphism is not the restriction of a q-c homeomorphism:For any two satellite copies M and M the induced Douady–Hubbardhomeomorphism is not q-c if the root multipliers λ = e2πip/q and λ = e2πip/q have q = q.",
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On quasi-conformal (in-) compatibility of satellite copies of the Mandelbrot set: I. / Lomonaco, Luna; Petersen, Carsten Lunde.

I: Inventiones Mathematicae, Bind 210, Nr. 2, 2017, s. 615-644.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - On quasi-conformal (in-) compatibility of satellite copies of the Mandelbrot set: I

AU - Lomonaco, Luna

AU - Petersen, Carsten Lunde

PY - 2017

Y1 - 2017

N2 - Douady and Hubbard (Ann Sci Ec Norm Suppl 4 18(2):287–343,1985) introduced the notion of polynomial-like maps. They used it to identifyhomeomorphic copies M of the Mandelbrot set inside the Mandelbrotset M. These copies can be primitive (with a root cusp) or satellite (withouta root cusp). They conjectured that the primitive copies are quasiconformallyhomeomorphic to M, and that the satellite ones are quasiconformallyhomeomorphic to M outside any small neighbourhood of the root. These conjectures are now Theorems due to Lyubich (Ann Math 149:319–420, 1999).The satellite copies Mp/q are clearly not q-c homeomorphic to M. But arethey mutually q-c homeomorphic? Or even q-c homeomorphic to half of thelogistic Mandelbrot set? In this paper we prove that, in general, the inducedDouady–Hubbard homeomorphism is not the restriction of a q-c homeomorphism:For any two satellite copies M and M the induced Douady–Hubbardhomeomorphism is not q-c if the root multipliers λ = e2πip/q and λ = e2πip/q have q = q.

AB - Douady and Hubbard (Ann Sci Ec Norm Suppl 4 18(2):287–343,1985) introduced the notion of polynomial-like maps. They used it to identifyhomeomorphic copies M of the Mandelbrot set inside the Mandelbrotset M. These copies can be primitive (with a root cusp) or satellite (withouta root cusp). They conjectured that the primitive copies are quasiconformallyhomeomorphic to M, and that the satellite ones are quasiconformallyhomeomorphic to M outside any small neighbourhood of the root. These conjectures are now Theorems due to Lyubich (Ann Math 149:319–420, 1999).The satellite copies Mp/q are clearly not q-c homeomorphic to M. But arethey mutually q-c homeomorphic? Or even q-c homeomorphic to half of thelogistic Mandelbrot set? In this paper we prove that, in general, the inducedDouady–Hubbard homeomorphism is not the restriction of a q-c homeomorphism:For any two satellite copies M and M the induced Douady–Hubbardhomeomorphism is not q-c if the root multipliers λ = e2πip/q and λ = e2πip/q have q = q.

KW - Mandelbrotmængden

UR - https://arxiv.org/pdf/1505.05422.pdf

U2 - 10.1007/s00222-017-0737-1

DO - 10.1007/s00222-017-0737-1

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JO - Inventiones Mathematicae

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