Abstract
We show that there exist polynomial endomorphisms of $\mathbb{C}^2$, possessing a wandering Fatou component. These mappings are polynomial skew-products, and can be chosen to extend holomorphically of $\mathbb{P}^2(\mathbb{C})$. We also find real examples with wandering domains in $\mathbb{R}^2$. The proof relies on parabolic implosion techniques and is based on an original idea of M. Lyubich.
-
[baker] I. N. Baker, "An entire function which has wandering domains," J. Austral. Math. Soc. Ser. A, vol. 22, iss. 2, pp. 173-176, 1976.
@article{baker, mrkey = {0419759},
author = {Baker, I. N.},
title = {An entire function which has wandering domains},
journal = {J. Austral. Math. Soc. Ser. A},
volume = {22},
year = {1976},
number = {2},
pages = {173--176},
mrclass = {30A66},
mrnumber = {0419759},
mrreviewer = {L. S. O. Liverpool},
zblnumber = {0335.30001},
} -
[benedetto]
R. L. Benedetto, "Wandering domains and nontrivial reduction in non-Archimedean dynamics," Illinois J. Math., vol. 49, iss. 1, pp. 167-193, 2005.
@article{benedetto, mrkey = {2157374},
author = {Benedetto, Robert L.},
title = {Wandering domains and nontrivial reduction in non-{A}rchimedean dynamics},
journal = {Illinois J. Math.},
fjournal = {Illinois Journal of Mathematics},
volume = {49},
year = {2005},
number = {1},
pages = {167--193},
issn = {0019-2082},
coden = {IJMTAW},
mrclass = {11S85 (37B99 37F10 37F50)},
mrnumber = {2157374},
mrreviewer = {Charles Favre},
url = {http://projecteuclid.org/euclid.ijm/1258138313},
zblnumber = {02212233},
} -
[bsu] E. Bedford, J. Smillie, and T. Ueda, Parabolic bifurcations in complex dimension two, 2012.
@misc{bsu,
author = {Bedford, E. and Smillie, J. and Ueda, T.},
title = {Parabolic bifurcations in complex dimension two},
year = {2012},
arxiv = {1208.2577},
} -
[bishop]
C. J. Bishop, "Constructing entire functions by quasiconformal folding," Acta Math., vol. 214, iss. 1, pp. 1-60, 2015.
@article{bishop, mrkey = {3316755},
author = {Bishop, Christopher J.},
title = {Constructing entire functions by quasiconformal folding},
journal = {Acta Math.},
fjournal = {Acta Mathematica},
volume = {214},
year = {2015},
number = {1},
pages = {1--60},
issn = {0001-5962},
mrclass = {30C65 (05C05)},
mrnumber = {3316755},
doi = {10.1007/s11511-015-0122-0},
zblnumber = {06428137},
} -
[bee]
X. Buff, J. Écalle, and A. Epstein, "Limits of degenerate parabolic quadratic rational maps," Geom. Funct. Anal., vol. 23, iss. 1, pp. 42-95, 2013.
@article{bee, mrkey = {3037897},
author = {Buff, Xavier and {É}calle, Jean and Epstein, Adam},
title = {Limits of degenerate parabolic quadratic rational maps},
journal = {Geom. Funct. Anal.},
fjournal = {Geometric and Functional Analysis},
volume = {23},
year = {2013},
number = {1},
pages = {42--95},
issn = {1016-443X},
mrclass = {37F10},
mrnumber = {3037897},
mrreviewer = {Alfredo Poirier},
doi = {10.1007/s00039-012-0203-6},
zblnumber = {06183906},
} -
[blokhlyubich]
A. M. Blokh and Y. M. Lyubich, "Non-existence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems 2. The smooth case," Ergodic Theory Dynam. Systems, vol. 9, iss. 4, pp. 751-758, 1989.
@article{blokhlyubich, mrkey = {1036906},
author = {Blokh, A. M. and Lyubich, M. Yu.},
title = {Non-existence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems 2. {T}he smooth case},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {9},
year = {1989},
number = {4},
pages = {751--758},
issn = {0143-3857},
mrclass = {58F12},
mrnumber = {1036906},
mrreviewer = {J. Angel Rodr{\'ı}guez},
doi = {10.1017/S0143385700005319},
zblnumber = {0665.58024},
} -
[boc-thalerfornaesspeters]
L. Boc-Thaler, J. E. Fornaess, and H. Peters, "Fatou components with punctured limit sets," Ergodic Theory Dynam. Systems, vol. 35, iss. 5, pp. 1380-1393, 2015.
@article{boc-thalerfornaesspeters, mrkey = {3365727},
author = {Boc-Thaler, Luka and Forn{æ}ss, John Erik and Peters, Han},
title = {Fatou components with punctured limit sets},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {35},
year = {2015},
number = {5},
pages = {1380--1393},
issn = {0143-3857},
mrclass = {32H50 (37F50)},
mrnumber = {3365727},
doi = {10.1017/etds.2013.115},
zblnumber = {06495615},
} -
[denjoy] A. Denjoy, "Sur les courbes définies par les équations différentielles à la surface du tore," J. Math. Pures Appl., vol. 11, pp. 333-376, 1932.
@article{denjoy,
author = {Denjoy, A.},
title = {Sur les courbes définies par les équations différentielles à la surface du tore},
journal = {J. Math. Pures Appl.},
volume = {11},
year = {1932},
pages = {333--376},
zblnumber = {0006.30501},
} -
[douady]
A. Douady, "Does a Julia set depend continuously on the polynomial?," in Complex Dynamical Systems, Providence, RI: Amer. Math. Soc., 1994, vol. 49, pp. 91-138.
@incollection{douady, mrkey = {1315535},
author = {Douady, Adrien},
title = {Does a {J}ulia set depend continuously on the polynomial?},
booktitle = {Complex Dynamical Systems},
venue = {{C}incinnati, {OH},
1994},
series = {Proc. Sympos. Appl. Math.},
volume = {49},
pages = {91--138},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {1994},
mrclass = {58F23 (30D05)},
mrnumber = {1315535},
doi = {10.1090/psapm/049/1315535},
zblnumber = {0934.30023},
} -
@misc{dujardin,
author = {Dujardin, Romain},
title = {A non-laminar dynamical {G}reen current},
note = {{\em Math. Ann.},
online first: 31 July 2015},
doi = {10.1007/s00208-015-1274-0},
sortyear={2015},
key={Duj15},
} -
[dl]
R. Dujardin and M. Lyubich, "Stability and bifurcations for dissipative polynomial automorphisms of $\Bbb{C}^2$," Invent. Math., vol. 200, iss. 2, pp. 439-511, 2015.
@article{dl, mrkey = {3338008},
author = {Dujardin, Romain and Lyubich, Mikhail},
title = {Stability and bifurcations for dissipative polynomial automorphisms of {$\Bbb{C}\sp 2$}},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {200},
year = {2015},
number = {2},
pages = {439--511},
issn = {0020-9910},
mrclass = {37F45},
mrnumber = {3338008},
doi = {10.1007/s00222-014-0535-y},
zblnumber = {06442709},
} -
[Ec] J. Écalle, Les Fonctions Résurgentes. Tome III: L’Équation du Pont et la Classification Analytique des Objects Locaux, Orsay: Université de Paris-Sud, Département de Mathématiques, 1985, vol. 85.
@book{Ec, mrkey = {0852210},
author = {{É}calle, Jean},
title = {Les Fonctions Résurgentes. {T}ome {III}: L'{É}quation du Pont et la Classification Analytique des Objects Locaux},
series = {Publications Mathématiques d'Orsay [Mathematical Publications of Orsay]},
volume = {85},
publisher = {Université de Paris-Sud, Département de Mathématiques},
address = {Orsay},
year = {1985},
pages = {587},
mrclass = {32Bxx (30B40 32A10 34-02 58C07 58F08)},
mrnumber = {0852210},
mrreviewer = {Carlos A. Berenstein},
zblnumber = {0602.30029},
} -
@book{e, mrkey = {2690048},
author = {Epstein, Adam Lawrence},
title = {Towers of Finite Type Complex Analytic Maps},
note = {Thesis (Ph.D.)--City University of New York},
publisher = {ProQuest LLC},
address = {Ann Arbor, MI},
year = {1993},
pages = {160},
mrclass = {Thesis},
mrnumber = {2690048},
url = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9405521},
} -
[el1]
A. E. Eremenko and J. M. Lyubich, "Examples of entire functions with pathological dynamics," J. London Math. Soc., vol. 36, iss. 3, pp. 458-468, 1987.
@article{el1, mrkey = {0918638},
author = {Eremenko, A. E. and Lyubich, M. Ju.},
title = {Examples of entire functions with pathological dynamics},
journal = {J. London Math. Soc.},
fjournal = {Journal of the London Mathematical Society. Second Series},
volume = {36},
year = {1987},
number = {3},
pages = {458--468},
issn = {0024-6107},
coden = {JLMSAK},
mrclass = {30D05 (58F11)},
mrnumber = {0918638},
mrreviewer = {Janina Kotus},
doi = {10.1112/jlms/s2-36.3.458},
zblnumber = {0601.30033},
} -
[eremenkolyubich]
A. E. Eremenko and Y. M. Lyubich, "Dynamical properties of some classes of entire functions," Ann. Inst. Fourier $($Grenoble$)$, vol. 42, iss. 4, pp. 989-1020, 1992.
@article{eremenkolyubich, mrkey = {1196102},
author = {Eremenko, A. E. and Lyubich, M. Yu.},
title = {Dynamical properties of some classes of entire functions},
journal = {Ann. Inst. Fourier $($Grenoble$)$},
fjournal = {Université de Grenoble. Annales de l'Institut Fourier},
volume = {42},
year = {1992},
number = {4},
pages = {989--1020},
issn = {0373-0956},
coden = {AIFUA7},
mrclass = {30D05 (58F23)},
mrnumber = {1196102},
mrreviewer = {Ben Bielefeld},
url = {http://www.numdam.org/item?id=AIF_1992__42_4_989_0},
zblnumber = {0735.58031},
} -
[fstranscendental]
J. E. Fornaess and N. Sibony, "Fatou and Julia sets for entire mappings in ${\bf C}^k$," Math. Ann., vol. 311, iss. 1, pp. 27-40, 1998.
@article{fstranscendental, mrkey = {1624255},
author = {Fornaess, John Erik and Sibony, Nessim},
title = {Fatou and {J}ulia sets for entire mappings in {${\bf C}\sp k$}},
journal = {Math. Ann.},
fjournal = {Mathematische Annalen},
volume = {311},
year = {1998},
number = {1},
pages = {27--40},
issn = {0025-5831},
coden = {MAANA},
mrclass = {32H50},
mrnumber = {1624255},
mrreviewer = {Marco Abate},
doi = {10.1007/s002080050174},
zblnumber = {0931.37016},
} -
[fs]
J. E. Fornaess and N. Sibony, "Some open problems in higher dimensional complex analysis and complex dynamics," Publ. Mat., vol. 45, iss. 2, pp. 529-547, 2001.
@article{fs, mrkey = {1876919},
author = {Forn{æ}ss, John Erik and Sibony, Nessim},
title = {Some open problems in higher dimensional complex analysis and complex dynamics},
journal = {Publ. Mat.},
fjournal = {Publicacions Matemàtiques},
volume = {45},
year = {2001},
number = {2},
pages = {529--547},
issn = {0214-1493},
mrclass = {32H50 (32E40 32W05 37Fxx)},
mrnumber = {1876919},
mrreviewer = {Marco Abate},
doi = {10.5565/PUBLMAT_45201_11},
zblnumber = {0993.32001},
} -
[gk]
L. R. Goldberg and L. Keen, "A finiteness theorem for a dynamical class of entire functions," Ergodic Theory Dynam. Systems, vol. 6, iss. 2, pp. 183-192, 1986.
@article{gk, mrkey = {0857196},
author = {Goldberg, Lisa R. and Keen, Linda},
title = {A finiteness theorem for a dynamical class of entire functions},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {6},
year = {1986},
number = {2},
pages = {183--192},
issn = {0143-3857},
mrclass = {58F99 (30D05 58F08)},
mrnumber = {0857196},
mrreviewer = {M. Rees},
doi = {10.1017/S0143385700003394},
zblnumber = {0657.58011},
} -
[guckenheimer]
J. Guckenheimer, "Sensitive dependence to initial conditions for one-dimensional maps," Comm. Math. Phys., vol. 70, iss. 2, pp. 133-160, 1979.
@article{guckenheimer, mrkey = {0553966},
author = {Guckenheimer, John},
title = {Sensitive dependence to initial conditions for one-dimensional maps},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {70},
year = {1979},
number = {2},
pages = {133--160},
issn = {0010-3616},
coden = {CMPHAY},
mrclass = {58F13 (34C35)},
mrnumber = {0553966},
mrreviewer = {Yu. E. Gliklikh},
doi = {10.1007/BF01982351},
zblnumber = {0429.58012},
} -
[h]
M. Hakim, "Analytic transformations of $(\Bbb C^p,0)$ tangent to the identity," Duke Math. J., vol. 92, iss. 2, pp. 403-428, 1998.
@article{h, mrkey = {1612730},
author = {Hakim, Monique},
title = {Analytic transformations of {$(\bold C\sp p,0)$} tangent to the identity},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {92},
year = {1998},
number = {2},
pages = {403--428},
issn = {0012-7094},
coden = {DUMJAO},
mrclass = {32H50 (32H02)},
mrnumber = {1612730},
mrreviewer = {Brendan J. Weickert},
doi = {10.1215/S0012-7094-98-09212-2},
zblnumber = {0952.32012},
} -
[herman]
M. Herman, "Exemples de fractions rationnelles ayant une orbite dense sur la sphère de Riemann," Bull. Soc. Math. France, vol. 112, iss. 1, pp. 93-142, 1984.
@article{herman, mrkey = {0771920},
author = {Herman, Michael-R.},
title = {Exemples de fractions rationnelles ayant une orbite dense sur la sphère de {R}iemann},
journal = {Bull. Soc. Math. France},
fjournal = {Bulletin de la Société Mathématique de France},
volume = {112},
year = {1984},
number = {1},
pages = {93--142},
issn = {0037-9484},
coden = {BSMFAA},
mrclass = {58F08 (30D05 58F11)},
mrnumber = {0771920},
mrreviewer = {I. N. Baker},
url = {http://www.numdam.org/item?id=BSMF_1984__112__93_0},
zblnumber = {0559.58020},
} -
[jonsson]
M. Jonsson, "Dynamics of polynomial skew products on $\Bbb C^2$," Math. Ann., vol. 314, iss. 3, pp. 403-447, 1999.
@article{jonsson, mrkey = {1704543},
author = {Jonsson, Mattias},
title = {Dynamics of polynomial skew products on {$\bold C\sp 2$}},
journal = {Math. Ann.},
fjournal = {Mathematische Annalen},
volume = {314},
year = {1999},
number = {3},
pages = {403--447},
issn = {0025-5831},
coden = {MAANA},
mrclass = {32H50 (37F10 37F15)},
mrnumber = {1704543},
mrreviewer = {Jean-Yves Briend},
doi = {10.1007/s002080050301},
zblnumber = {0940.37018},
} -
[lavaurs] P. Lavaurs, Systèmes dynamiques holomorphiques : explosion des points périodiques, 1989.
@misc{lavaurs,
author = {Lavaurs, P.},
title = {Systèmes dynamiques holomorphiques : explosion des points périodiques},
note = {Ph. D. Thesis, Université Paris-Sud},
year = {1989},
} -
@book{lilov, mrkey = {2706157},
author = {Lilov, Krastio},
title = {Fatou theory in two dimensions},
note = {Thesis (Ph.D.)--University of Michigan},
publisher = {ProQuest LLC, Ann Arbor, MI},
year = {2004},
pages = {111},
isbn = {978-0496-85046-4},
mrclass = {Thesis},
mrnumber = {2706157},
url = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3137876},
} -
[lyubich]
Y. M. Lyubich, "Non-existence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems. I. The case of negative Schwarzian derivative," Ergodic Theory Dynam. Systems, vol. 9, iss. 4, pp. 737-749, 1989.
@article{lyubich, mrkey = {1036905},
author = {Lyubich, M. Yu.},
title = {Non-existence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems. {I}. {T}he case of negative {S}chwarzian derivative},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {9},
year = {1989},
number = {4},
pages = {737--749},
issn = {0143-3857},
mrclass = {58F12},
mrnumber = {1036905},
mrreviewer = {J. Angel Rodr{\'ı}guez},
doi = {10.1017/S0143385700005307},
zblnumber = {0665.58023},
} -
[lyubichpeters]
M. Lyubich and H. Peters, "Classification of invariant Fatou components for dissipative Hénon maps," Geom. Funct. Anal., vol. 24, iss. 3, pp. 887-915, 2014.
@article{lyubichpeters, mrkey = {3213832},
author = {Lyubich, Mikhail and Peters, Han},
title = {Classification of invariant {F}atou components for dissipative {H}énon maps},
journal = {Geom. Funct. Anal.},
fjournal = {Geometric and Functional Analysis},
volume = {24},
year = {2014},
number = {3},
pages = {887--915},
issn = {1016-443X},
mrclass = {32H50 (37F50)},
mrnumber = {3213832},
mrreviewer = {Roland K. W. Roeder},
doi = {10.1007/s00039-014-0280-9},
zblnumber = {1301.32013},
} -
[martensmelostrien]
M. Martens, W. de Melo, and S. van Strien, "Julia-Fatou-Sullivan theory for real one-dimensional dynamics," Acta Math., vol. 168, iss. 3-4, pp. 273-318, 1992.
@article{martensmelostrien, mrkey = {1161268},
author = {Martens, M. and de Melo, W. and van Strien, S.},
title = {Julia-{F}atou-{S}ullivan theory for real one-dimensional dynamics},
journal = {Acta Math.},
fjournal = {Acta Mathematica},
volume = {168},
year = {1992},
number = {3-4},
pages = {273--318},
issn = {0001-5962},
coden = {ACMAA8},
mrclass = {58F23 (58F12)},
mrnumber = {1161268},
mrreviewer = {P. Mendes},
doi = {10.1007/BF02392981},
zblnumber ={0761.58007},
} -
@misc{petersvivas,
author = {Peters, H. and Vivas, L.},
title = {Polynomial skew-products with wandering {F}atou-disks},
note={{\em Math. Zeitschrift},
online first: 14 January 2016, 18pp.},
doi={10.1007/s00209-015-1600-y},
sortyear = {2016},
key={PV16},
} -
[shishikura] M. Shishikura, "Bifurcation of parabolic fixed points," in The Mandelbrot Set, Theme and Variations, Cambridge Univ. Press, Cambridge, 2000, vol. 274, pp. 325-363.
@incollection{shishikura, mrkey = {1765097},
author = {Shishikura, Mitsuhiro},
title = {Bifurcation of parabolic fixed points},
booktitle = {The {M}andelbrot Set, Theme and Variations},
series = {London Math. Soc. Lecture Note Ser.},
volume = {274},
pages = {325--363},
publisher = {Cambridge Univ. Press, Cambridge},
year = {2000},
mrclass = {37F45 (30D05)},
mrnumber = {1765097},
mrreviewer = {Carsten Lunde Petersen},
zblnumber = {1062.37043},
} -
[sul]
D. Sullivan, "Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains," Ann. of Math., vol. 122, iss. 3, pp. 401-418, 1985.
@article{sul, mrkey = {0819553},
author = {Sullivan, Dennis},
title = {Quasiconformal homeomorphisms and dynamics. {I}. {S}olution of the {F}atou-{J}ulia problem on wandering domains},
journal = {Ann. of Math.},
fjournal = {Annals of Mathematics. Second Series},
volume = {122},
year = {1985},
number = {3},
pages = {401--418},
issn = {0003-486X},
coden = {ANMAAH},
mrclass = {58F11 (20H05 30C60 30D05 57R99 58F08)},
mrnumber = {0819553},
mrreviewer = {L. Keen},
doi = {10.2307/1971308},
zblnumber = {0589.30022},
} -
[trucco]
E. Trucco, "Wandering Fatou components and algebraic Julia sets," Bull. Soc. Math. France, vol. 142, iss. 3, pp. 411-464, 2014.
@article{trucco, mrkey = {3295719},
author = {Trucco, Eugenio},
title = {Wandering {F}atou components and algebraic {J}ulia sets},
journal = {Bull. Soc. Math. France},
fjournal = {Bulletin de la Société Mathématique de France},
volume = {142},
year = {2014},
number = {3},
pages = {411--464},
issn = {0037-9484},
mrclass = {37P40 (37F50 37P05 37P50)},
mrnumber = {3295719},
mrreviewer = {Y{û}suke Okuyama},
zblnumber = {06412749},
URL = {http://smf4.emath.fr/Publications/Bulletin/142/html/smf_bull_142_411-464.php},
}