Comptes Rendus
Dynamical Systems
Wandering triangles exist
Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 365-370.

W.P. Thurston introduced closed σd-invariant laminations (where σd=zd:S1S1, d2) as a tool in complex dynamics. He defined wandering triangles as triples TS1 such that σdn(T) consists of three distinct points for all n0 and the convex hulls of all the sets σdn(T) in the plane are pairwise disjoint, and proved that σ2 admits no wandering triangles. We show that for every d3 there exist uncountably many σd-invariant closed laminations with wandering triangles and pairwise non-conjugate factor maps of σd on the corresponding quotient spaces.

Les laminations fermées σd-invariantes (où σd=zd:S1S1,d2) ont été introduites par W. P. Thurston comme un outil pour l'étude des systèmes dynamiques dans le plan complexe. Il avait défini les triangles errants comme étant des triplets TS1 tels que σdn(T) est composé des trois points distincts pour tout n0, et les enveloppes convexes de tous les ensembles σdn(T) sont deux-à-deux disjointes dans le plan complexe. Il avait démontré que σ2 n'admet pas des triangles errants. Nous montrons que pour tout d3 il existe une collection nondénombrable de laminations fermées σd-invariantes qui ont des triangles errants et des applications-facteurs de σd non-conjuguées, deux-à-deux distinctes, sur les espaces quotients associés.

Published online:
DOI: 10.1016/j.crma.2004.06.024
Alexander Blokh 1; Lex Oversteegen 1

1 Department of Mathematics, UAB, Birmingham, AL 35294, USA
     author = {Alexander Blokh and Lex Oversteegen},
     title = {Wandering triangles exist},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {365--370},
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     doi = {10.1016/j.crma.2004.06.024},
     language = {en},
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%A Lex Oversteegen
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%J Comptes Rendus. Mathématique
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Alexander Blokh; Lex Oversteegen. Wandering triangles exist. Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 365-370. doi : 10.1016/j.crma.2004.06.024.

[1] A. Blokh; G. Levin An inequality for laminations, Julia sets and “growing trees”, Ergodic Theory Dynamical Systems, Volume 22 (2002), pp. 63-97

[2] A. Blokh; G. Levin On dynamics of vertices of locally connected polynomial Julia sets, Proc. Amer. Math. Soc., Volume 130 (2002), pp. 3219-3230

[3] B. Branner; J. Hubbard The iteration of cubic polynomials. Part I: The global topology of parameter space, Acta Math., Volume 160 (1988), pp. 143-206

[4] A. Douady; J.H. Hubbard Étude dynamique des polynômes complexes I, Publ. Math. Orsay, Volume 84-02 (1984)

[5] A. Douady; J.H. Hubbard Étude dynamique des polynômes complexes II, Publ. Math. Orsay, Volume 85-04 (1985)

[6] J. Kiwi Wandering orbit portraits, Trans. Amer. Math. Soc., Volume 354 (2002), pp. 1473-1485

[7] J. Kiwi, Real laminations and the topological dynamics of complex polynomials, Adv. in Math., in press

[8] G. Levin On backward stability of holomorphic dynamical systems, Fund. Math., Volume 158 (1998), pp. 97-107

[9] L. Oversteegen; J. Rogers An inverse limit description of an atriodic tree-like continuum and an induced map without a fixed point, Houston J. Math, Volume 6 (1980), pp. 549-564

[10] W. Thurston, The combinatorics of iterated rational maps, Preprint, 1985

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