Abstract
We show that in a generic finite-dimensional real-analytic family of real-analytic multimodal maps, the subset of parameters on which the corresponding map has a solenoidal attractor with bounded combinatorics is a set with zero Lebesgue measure.
-
[a_book]
L. V. Ahlfors, Lectures on Quasiconformal Mappings, Second ed., Amer. Math. Soc., Providence, RI, 2006, vol. 38.
@BOOK{a_book,
author = {Ahlfors, Lars V.},
title = {Lectures on Quasiconformal Mappings},
series = {Univ. Lecture Ser.},
volume = {38},
edition = {Second},
note = {with supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2006},
pages = {viii+162},
isbn = {0-8218-3644-7},
mrclass = {30-01 (30-02 30C62 30D05 30F45 30F60)},
mrnumber = {2241787},
mrreviewer = {Edward Crane},
doi = {10.1090/ulect/038},
url = {https://doi.org/10.1090/ulect/038},
zblnumber = {1103.30001},
} -
[al33]
A. Avila and M. Lyubich, "The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes," Publ. Math. Inst. Hautes Études Sci., iss. 114, pp. 171-223, 2011.
@ARTICLE{al33,
author = {Avila, Artur and Lyubich, Mikhail},
title = {The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes},
journal = {Publ. Math. Inst. Hautes \'{E}tudes Sci.},
fjournal = {Publications Mathématiques. Institut de Hautes \'{E}tudes Scientifiques},
number = {114},
year = {2011},
pages = {171--223},
issn = {0073-8301},
mrclass = {37E20 (37E05 37F25)},
mrnumber = {2854860},
mrreviewer = {Henk Bruin},
doi = {10.1007/s10240-011-0034-2},
url = {https://doi.org/10.1007/s10240-011-0034-2},
zblnumber = {1286.37047},
} -
[alm]
A. Avila, M. Lyubich, and W. de Melo, "Regular or stochastic dynamics in real analytic families of unimodal maps," Invent. Math., vol. 154, iss. 3, pp. 451-550, 2003.
@ARTICLE{alm,
author = {Avila, Artur and Lyubich, Mikhail and de Melo, Welington},
title = {Regular or stochastic dynamics in real analytic families of unimodal maps},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {154},
year = {2003},
number = {3},
pages = {451--550},
issn = {0020-9910},
mrclass = {37E05 (30D05 37C15 37E20 37F45)},
mrnumber = {2018784},
mrreviewer = {Feliks Przytycki},
doi = {10.1007/s00222-003-0307-6},
url = {https://doi.org/10.1007/s00222-003-0307-6},
zblnumber = {1050.37018},
} -
[als]
A. Avila, M. Lyubich, and W. Shen, "Parapuzzle of the Multibrot set and typical dynamics of unimodal maps," J. Eur. Math. Soc. (JEMS), vol. 13, iss. 1, pp. 27-56, 2011.
@ARTICLE{als,
author = {Avila, Artur and Lyubich, Mikhail and Shen, Weixiao},
title = {Parapuzzle of the {M}ultibrot set and typical dynamics of unimodal maps},
journal = {J. Eur. Math. Soc. (JEMS)},
fjournal = {Journal of the European Mathematical Society (JEMS)},
volume = {13},
year = {2011},
number = {1},
pages = {27--56},
issn = {1435-9855},
mrclass = {37F45 (37C40 37E05 37F10 37F35)},
mrnumber = {2735075},
mrreviewer = {Henk Bruin},
doi = {10.4171/JEMS/243},
url = {https://doi.org/10.4171/JEMS/243},
zblnumber = {1213.37076},
} -
[am2]
A. Avila and C. G. Moreira, "Phase-parameter relation and sharp statistical properties for general families of unimodal maps," in Geometry and dDynamics, Amer. Math. Soc., Providence, RI, 2005, vol. 389, pp. 1-42.
@INCOLLECTION{am2,
author = {Avila, Artur and Moreira, Carlos Gustavo},
title = {Phase-parameter relation and sharp statistical properties for general families of unimodal maps},
booktitle = {Geometry and dDynamics},
series = {Contemp. Math.},
volume = {389},
pages = {1--42},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2005},
mrclass = {37E05 (37E20 37F25 82C05)},
mrnumber = {2181956},
mrreviewer = {Masato Tsujii},
doi = {10.1090/conm/389/07270},
url = {https://doi.org/10.1090/conm/389/07270},
zblnumber = {1145.37022},
} -
[bl]
A. M. Blokh and Y. M. Lyubich, "Attractors of the transformations of the interval," Funktsional. Anal. i Prilozhen., vol. 21, iss. 2, pp. 70-71, 1987.
@ARTICLE{bl,
author = {Blokh, A. M. and Lyubich, M. Yu.},
title = {Attractors of the transformations of the interval},
journal = {Funktsional. Anal. i Prilozhen.},
fjournal = {Akademiya Nauk SSSR. Funktsional\cprime nyĭ Analiz i ego Prilozheniya},
volume = {21},
year = {1987},
number = {2},
pages = {70--71},
issn = {0374-1990},
mrclass = {58F12 (34C35 58F11)},
mrnumber = {0902297},
mrreviewer = {Zuo Ling Zhou},
zblnumber = {0653.58022},
url = {http://mi.mathnet.ru/eng/faa/v21/i2/p70},
} -
[bl2]
A. M. Blokh and Y. M. Lyubich, "Decomposition of one-dimensional dynamical systems into ergodic components. The case of a negative Schwarzian derivative," Algebra i Analiz, vol. 1, iss. 1, pp. 128-145, 1989.
@ARTICLE{bl2,
author = {Blokh, A. M. and Lyubich, M. Yu.},
title = {Decomposition of one-dimensional dynamical systems into ergodic components. {T}he case of a negative {S}chwarzian derivative},
journal = {Algebra i Analiz},
fjournal = {Algebra i Analiz},
volume = {1},
year = {1989},
number = {1},
pages = {128--145},
issn = {0234-0852},
mrclass = {58F11 (58F12)},
mrnumber = {1015337},
mrreviewer = {Feliks Przytycki},
zblnumber = {0718.58024},
url = {http://mi.mathnet.ru/eng/aa/v1/i1/p128},
} -
[wild2]
H. Bruin, G. Keller, T. Nowicki, and S. van Strien, "Wild Cantor attractors exist," Ann. of Math. (2), vol. 143, iss. 1, pp. 97-130, 1996.
@ARTICLE{wild2,
author = {Bruin, H. and Keller, G. and Nowicki, T. and van Strien, S.},
title = {Wild {C}antor attractors exist},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {143},
year = {1996},
number = {1},
pages = {97--130},
issn = {0003-486X},
mrclass = {58F12 (30D05 58F23)},
mrnumber = {1370759},
mrreviewer = {Hartje Kriete},
doi = {10.2307/2118654},
url = {https://doi.org/10.2307/2118654},
zblnumber = {0848.58016},
} -
[wild4]
H. Bruin, G. Keller, and M. St. Pierre, "Adding machines and wild attractors," Ergodic Theory Dynam. Systems, vol. 17, iss. 6, pp. 1267-1287, 1997.
@ARTICLE{wild4,
author = {Bruin, Henk and Keller, Gerhard and St. Pierre, Matthias},
title = {Adding machines and wild attractors},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {17},
year = {1997},
number = {6},
pages = {1267--1287},
issn = {0143-3857},
mrclass = {58F12 (11A63 58F13)},
mrnumber = {1488317},
mrreviewer = {Peter J. Grabner},
doi = {10.1017/S0143385797086392},
url = {https://doi.org/10.1017/S0143385797086392},
zblnumber = {0898.58012},
} -
[sbr]
H. Bruin, W. Shen, and S. van Strien, "Existence of unique SRB-measures is typical for real unicritical polynomial families," Ann. Sci. École Norm. Sup. (4), vol. 39, iss. 3, pp. 381-414, 2006.
@ARTICLE{sbr,
author = {Bruin, Henk and Shen, Weixiao and van Strien, Sebastian},
title = {Existence of unique {SRB}-measures is typical for real unicritical polynomial families},
journal = {Ann. Sci. \'{E}cole Norm. Sup. (4)},
fjournal = {Annales Scientifiques de l'\'{E}cole Normale Supérieure. Quatrième Série},
volume = {39},
year = {2006},
number = {3},
pages = {381--414},
issn = {0012-9593},
mrclass = {37E05 (37C40 37D99 37F10)},
mrnumber = {2265674},
mrreviewer = {Mike Todd},
doi = {10.1016/j.ansens.2006.02.001},
url = {https://doi.org/10.1016/j.ansens.2006.02.001},
zblnumber = {1112.37018},
} -
[clark]
T. Clark, "Regular or stochastic dynamics in families of higher-degree unimodal maps," Ergodic Theory Dynam. Systems, vol. 34, iss. 5, pp. 1538-1566, 2014.
@ARTICLE{clark,
author = {Clark, Trevor},
title = {Regular or stochastic dynamics in families of higher-degree unimodal maps},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {34},
year = {2014},
number = {5},
pages = {1538--1566},
issn = {0143-3857},
mrclass = {37E05 (37E20 37F25)},
mrnumber = {3255432},
doi = {10.1017/etds.2013.10},
url = {https://doi.org/10.1017/etds.2013.10},
zblnumber = {1322.37019},
} -
[fma]
E. de Faria, W. de Melo, and A. Pinto, "Global hyperbolicity of renormalization for $C^r$ unimodal mappings," Ann. of Math. (2), vol. 164, iss. 3, pp. 731-824, 2006.
@ARTICLE{fma,
author = {de Faria, Edson and de Melo, Welington and Pinto, Alberto},
title = {Global hyperbolicity of renormalization for {$C^r$} unimodal mappings},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {164},
year = {2006},
number = {3},
pages = {731--824},
issn = {0003-486X},
mrclass = {37E05 (37D20 37E20)},
mrnumber = {2259245},
mrreviewer = {Henk Bruin},
doi = {10.4007/annals.2006.164.731},
url = {https://doi.org/10.4007/annals.2006.164.731},
zblnumber = {1129.37021},
} -
[ms]
W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer-Verlag, Berlin, 1993, vol. 25.
@BOOK{ms,
author = {de Melo, Welington and van Strien, Sebastian},
title = {One-Dimensional Dynamics},
series = {Ergeb. Math. Grenzgeb.},
volume = {25},
publisher = {Springer-Verlag, Berlin},
year = {1993},
pages = {xiv+605},
isbn = {3-540-56412-8},
mrclass = {58F03 (58-02 58Fxx)},
mrnumber = {1239171},
mrreviewer = {Feliks Przytycki},
doi = {10.1007/978-3-642-78043-1},
url = {https://doi.org/10.1007/978-3-642-78043-1},
zblnumber = {0791.58003},
} -
[dgp]
B. Derrida, A. Gervois, and Y. Pomeau, "Universal metric properties of bifurcations of endomorphisms," J. Phys. A, vol. 12, iss. 3, pp. 269-296, 1979.
@ARTICLE{dgp,
author = {Derrida, B. and Gervois, A. and Pomeau, Y.},
title = {Universal metric properties of bifurcations of endomorphisms},
journal = {J. Phys. A},
fjournal = {Journal of Physics. A. Mathematical and General},
volume = {12},
year = {1979},
number = {3},
pages = {269--296},
issn = {0305-4470},
mrclass = {58F14 (49A99 58F13)},
mrnumber = {0524170},
mrreviewer = {L. A. Bunimovich},
doi = {10.1088/0305-4470/12/3/004},
url = {https://doi.org/10.1088/0305-4470/12/3/004},
zblnumber = {0416.28011},
} -
[fei1]
M. J. Feigenbaum, "Quantitative universality for a class of nonlinear transformations," J. Statist. Phys., vol. 19, iss. 1, pp. 25-52, 1978.
@ARTICLE{fei1,
author = {Feigenbaum, Mitchell J.},
title = {Quantitative universality for a class of nonlinear transformations},
journal = {J. Statist. Phys.},
fjournal = {Journal of Statistical Physics},
volume = {19},
year = {1978},
number = {1},
pages = {25--52},
issn = {0022-4715},
mrclass = {58F20 (39A12 65Q05)},
mrnumber = {0501179},
mrreviewer = {Eugene Allgower},
doi = {10.1007/BF01020332},
url = {https://doi.org/10.1007/BF01020332},
zblnumber = {0509.58037},
} -
[fei2]
M. J. Feigenbaum, "The universal metric properties of nonlinear transformations," J. Statist. Phys., vol. 21, iss. 6, pp. 669-706, 1979.
@ARTICLE{fei2,
author = {Feigenbaum, Mitchell J.},
title = {The universal metric properties of nonlinear transformations},
journal = {J. Statist. Phys.},
fjournal = {Journal of Statistical Physics},
volume = {21},
year = {1979},
number = {6},
pages = {669--706},
issn = {0022-4715},
mrclass = {58F14},
mrnumber = {0555919},
mrreviewer = {Frederick R. Marotto},
doi = {10.1007/BF01107909},
url = {https://doi.org/10.1007/BF01107909},
zblnumber = {0515.58028},
} -
[sinai]
A. I. Golcprimeberg, . Y. G. Sinaui, and K. M. Khanin, "Universal properties of sequences of period-tripling bifurcations," Uspekhi Mat. Nauk, vol. 38, iss. 1(229), pp. 159-160, 1983.
@ARTICLE{sinai,
author = {Gol{\cprime}berg, A. I. and Sinaĭ,
{\relax Ya} G. and Khanin, K. M.},
title = {Universal properties of sequences of period-tripling bifurcations},
journal = {Uspekhi Mat. Nauk},
fjournal = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
volume = {38},
year = {1983},
number = {1(229)},
pages = {159--160},
issn = {0042-1316},
mrclass = {58F14 (58F08 58F30)},
mrnumber = {0693727},
mrreviewer = {Igor Gumowski},
zblnumber = {0534.30032},
doi = {10.1070/RM1983v038n01ABEH003398},
url = {https://doi.org/10.1070/RM1983v038n01ABEH003398},
} -
[gs]
J. Graczyk and G. Światek, "Generic hyperbolicity in the logistic family," Ann. of Math. (2), vol. 146, iss. 1, pp. 1-52, 1997.
@ARTICLE{gs,
author = {Graczyk, Jacek and \'{S}wiatek, Grzegorz},
title = {Generic hyperbolicity in the logistic family},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {146},
year = {1997},
number = {1},
pages = {1--52},
issn = {0003-486X},
mrclass = {58F03 (30C10 30C62 58F23)},
mrnumber = {1469316},
mrreviewer = {Feliks Przytycki},
doi = {10.2307/2951831},
url = {https://doi.org/10.2307/2951831},
zblnumber = {0936.37015},
} -
[hu]
J. Hu, Renormalization, Rigidity, and Universality in Bifurcation Theory, ProQuest LLC, Ann Arbor, MI, 1995.
@BOOK{hu,
author = {Hu, Jun},
title = {Renormalization, Rigidity, and Universality in Bifurcation Theory},
note = {thesis (Ph.D.)--City Univ. of New York},
publisher = {ProQuest LLC, Ann Arbor, MI},
year = {1995},
pages = {156},
mrclass = {Thesis},
mrnumber = {2693444},
url = {https://search.proquest.com/docview/304170974?accountid=13314},
zblnumber = {},
} -
[jak]
M. V. Jakobson, "Absolutely continuous invariant measures for one-parameter families of one-dimensional maps," Comm. Math. Phys., vol. 81, iss. 1, pp. 39-88, 1981.
@ARTICLE{jak,
author = {Jakobson, M. V.},
title = {Absolutely continuous invariant measures for one-parameter families of one-dimensional maps},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {81},
year = {1981},
number = {1},
pages = {39--88},
issn = {0010-3616},
mrclass = {58F11 (28D99 34C35 34C40)},
mrnumber = {0630331},
mrreviewer = {M. I. Brin},
doi = {10.1007/BF01941800},
url = {https://doi.org/10.1007/BF01941800},
zblnumber = {0497.58017},
} -
[kapo]
M. Kapovich, "On the dynamics of pseudo-Anosov homeomorphisms on representation varieties of surface groups," Ann. Acad. Sci. Fenn. Math., vol. 23, iss. 1, pp. 83-100, 1998.
@ARTICLE{kapo,
author = {Kapovich, Michael},
title = {On the dynamics of pseudo-{A}nosov homeomorphisms on representation varieties of surface groups},
journal = {Ann. Acad. Sci. Fenn. Math.},
fjournal = {Annales Academiæ Scientiarum Fennicæ . Mathematica},
volume = {23},
year = {1998},
number = {1},
pages = {83--100},
issn = {1239-629X},
mrclass = {57M50 (30C65 30F40 58F15)},
mrnumber = {1601847},
mrreviewer = {Feng Luo},
zblnumber = {0892.58058},
url = {https://www.acadsci.fi/mathematica/Vol23/kapovich.html},
} -
[wild3]
G. Keller and T. Nowicki, "Fibonacci maps re(al)visited," Ergodic Theory Dynam. Systems, vol. 15, iss. 1, pp. 99-120, 1995.
@ARTICLE{wild3,
author = {Keller, Gerhard and Nowicki, Tomasz},
title = {Fibonacci maps re(al)visited},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {15},
year = {1995},
number = {1},
pages = {99--120},
issn = {0143-3857},
mrclass = {58F12 (58F03)},
mrnumber = {1314971},
mrreviewer = {Remo Badii},
doi = {10.1017/S0143385700008269},
url = {https://doi.org/10.1017/S0143385700008269},
zblnumber = {0853.58072},
} -
[kss2]
O. Kozlovski, W. Shen, and S. van Strien, "Density of hyperbolicity in dimension one," Ann. of Math. (2), vol. 166, iss. 1, pp. 145-182, 2007.
@ARTICLE{kss2,
author = {Kozlovski, O. and Shen, W. and van Strien, S.},
title = {Density of hyperbolicity in dimension one},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {166},
year = {2007},
number = {1},
pages = {145--182},
issn = {0003-486X},
mrclass = {37E05 (37C20 37D20 37E10 37F30)},
mrnumber = {2342693},
mrreviewer = {Mike Todd},
doi = {10.4007/annals.2007.166.145},
url = {https://doi.org/10.4007/annals.2007.166.145},
zblnumber = {1138.37013},
} -
[kss1]
O. Kozlovski, W. Shen, and S. van Strien, "Rigidity for real polynomials," Ann. of Math. (2), vol. 165, iss. 3, pp. 749-841, 2007.
@ARTICLE{kss1,
author = {Kozlovski, O. and Shen, W. and van Strien, S.},
title = {Rigidity for real polynomials},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {165},
year = {2007},
number = {3},
pages = {749--841},
issn = {0003-486X},
mrclass = {37E05 (30C10 30C62 37C20 37E20 37F10 37F30 37F50)},
mrnumber = {2335796},
mrreviewer = {Henk Bruin},
doi = {10.4007/annals.2007.165.749},
url = {https://doi.org/10.4007/annals.2007.165.749},
zblnumber = {1129.37020},
} -
[lanford]
O. E. Lanford III, "A computer-assisted proof of the Feigenbaum conjectures," Bull. Amer. Math. Soc. (N.S.), vol. 6, iss. 3, pp. 427-434, 1982.
@ARTICLE{lanford,
author = {Lanford, III, Oscar E.},
title = {A computer-assisted proof of the {F}eigenbaum conjectures},
journal = {Bull. Amer. Math. Soc. (N.S.)},
fjournal = {Amer. Math. Soc.. Bulletin. New Series},
volume = {6},
year = {1982},
number = {3},
pages = {427--434},
issn = {0273-0979},
mrclass = {58F14 (70K50 82A05)},
mrnumber = {0648529},
doi = {10.1090/S0273-0979-1982-15008-X},
url = {https://doi.org/10.1090/S0273-0979-1982-15008-X},
zblnumber = {0487.58017},
} -
[lhyp]
M. Lyubich, "Dynamics of quadratic polynomials. I, II," Acta Math., vol. 178, iss. 2, pp. 185-297, 1997.
@ARTICLE{lhyp,
author = {Lyubich, Mikhail},
title = {Dynamics of quadratic polynomials. {I},
{II}},
journal = {Acta Math.},
fjournal = {Acta Mathematica},
volume = {178},
year = {1997},
number = {2},
pages = {185--297},
issn = {0001-5962},
mrclass = {58F23 (30C10 30D05)},
mrnumber = {1459261},
mrreviewer = {Grzegorz \'{S}wi\polhk atek},
doi = {10.1007/BF02392694},
url = {https://doi.org/10.1007/BF02392694},
zblnumber = {0908.58053},
} -
[lyu]
M. Lyubich, "Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture," Ann. of Math. (2), vol. 149, iss. 2, pp. 319-420, 1999.
@ARTICLE{lyu,
author = {Lyubich, Mikhail},
title = {Feigenbaum-{C}oullet-{T}resser universality and {M}ilnor's hairiness conjecture},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {149},
year = {1999},
number = {2},
pages = {319--420},
issn = {0003-486X},
mrclass = {37F25 (30D05 37E05 37E20 37F15 37F45)},
mrnumber = {1689333},
mrreviewer = {Welington de Melo},
doi = {10.2307/120968},
url = {https://doi.org/10.2307/120968},
zblnumber = {0945.37012},
} -
[lyu3] M. Lyubich, "Dynamics of quadratic polynomials. III. Parapuzzle and SBR measures," in Géométrie Complexe et Systèmes Dynamiques, Soc. Math. France, Paris, 2000, vol. 261, p. xii-xiii, 173.
@INCOLLECTION{lyu3,
author = {Lyubich, Mikhail},
title = {Dynamics of quadratic polynomials. {III}. {P}arapuzzle and {SBR} measures},
booktitle = {Géométrie Complexe et Systèmes Dynamiques},
venue = {Orsay, 1995},
series = {Astérisque},
volume = {261},
publisher = {Soc. Math. France, Paris},
year = {2000},
pages = {xii--xiii, 173--200},
issn = {0303-1179},
mrclass = {37F10 (37F15 37F25)},
mrnumber = {1755441},
zblnumber = {1044.37038},
} -
[lyuq]
M. Lyubich, "Almost every real quadratic map is either regular or stochastic," Ann. of Math. (2), vol. 156, iss. 1, pp. 1-78, 2002.
@ARTICLE{lyuq,
author = {Lyubich, Mikhail},
title = {Almost every real quadratic map is either regular or stochastic},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {156},
year = {2002},
number = {1},
pages = {1--78},
issn = {0003-486X},
mrclass = {37E05 (28D05 37A05 37E20 37F25)},
mrnumber = {1935840},
mrreviewer = {Michael Yampolsky},
doi = {10.2307/3597183},
url = {https://doi.org/10.2307/3597183},
zblnumber = {1160.37356},
} -
[mackay] R. S. MacKay and J. B. J. van Zeijts, "Period doubling for bimodal maps: a horseshoe for a renormalisation operator," Nonlinearity, vol. 1, iss. 1, pp. 253-277, 1988.
@ARTICLE{mackay,
author = {MacKay, R. S. and van Zeijts, J. B. J.},
title = {Period doubling for bimodal maps: a horseshoe for a renormalisation operator},
journal = {Nonlinearity},
fjournal = {Nonlinearity},
volume = {1},
year = {1988},
number = {1},
pages = {253--277},
issn = {0951-7715},
mrclass = {58F14 (58-04 82A25)},
mrnumber = {0928956},
mrreviewer = {Yong Nian Huang},
zblnumber = {0648.58028},
} -
[mc1]
C. T. McMullen, Complex Dynamics and Renormalization, Princeton Univ. Press, Princeton, NJ, 1994, vol. 135.
@BOOK{mc1,
author = {McMullen, Curtis T.},
title = {Complex Dynamics and Renormalization},
series = {Ann. of Math. Stud.},
volume = {135},
publisher = {Princeton Univ. Press, Princeton, NJ},
year = {1994},
pages = {x+214},
isbn = {0-691-02982-2; 0-691-02981-4},
mrclass = {58F23 (30D05)},
mrnumber = {1312365},
mrreviewer = {Gregery T. Buzzard},
zblnumber = {0822.30002},
doi = {10.1515/9781400882557},
url = {https://doi.org/10.1515/9781400882557},
} -
[mc2]
C. T. McMullen, Renormalization and 3-Manifolds which Fiber over the Circle, Princeton Univ. Press, Princeton, NJ, 1996, vol. 142.
@BOOK{mc2,
author = {McMullen, Curtis T.},
title = {Renormalization and 3-Manifolds which Fiber over the Circle},
series = {Ann. of Math. Stud.},
volume = {142},
publisher = {Princeton Univ. Press, Princeton, NJ},
year = {1996},
pages = {x+253},
isbn = {0-691-01154-0; 0-691-01153-2},
mrclass = {57N10 (26A18 30F40 58F99)},
mrnumber = {1401347},
mrreviewer = {Athanase Papadopoulos},
doi = {10.1515/9781400865178},
url = {https://doi.org/10.1515/9781400865178},
zblnumber = {0860.58002},
} -
[wild1]
J. Milnor, "On the concept of attractor," Comm. Math. Phys., vol. 99, iss. 2, pp. 177-195, 1985.
@ARTICLE{wild1,
author = {Milnor, John},
title = {On the concept of attractor},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {99},
year = {1985},
number = {2},
pages = {177--195},
issn = {0010-3616},
mrclass = {58F12 (58F08)},
mrnumber = {0790735},
mrreviewer = {Hans G. Bothe},
doi = {10.1007/BF01212280},
url = {https://doi.org/10.1007/BF01212280},
zblnumber = {0595.58028},
} -
[milnor]
J. Milnor, Dynamics in One Complex Variable, Third ed., Princeton Univ. Press, Princeton, NJ, 2006, vol. 160.
@BOOK{milnor,
author = {Milnor, John},
title = {Dynamics in One Complex Variable},
series = {Ann. of Math. Stud.},
volume = {160},
edition = {Third},
publisher = {Princeton Univ. Press, Princeton, NJ},
year = {2006},
pages = {viii+304},
isbn = {978-0-691-12488-9; 0-691-12488-4},
mrclass = {37Fxx (30-01 30D05 37-01)},
mrnumber = {2193309},
zblnumber = {1085.30002},
doi = {10.2307/j.ctt7rnxn},
url = {https://doi.org/10.2307/j.ctt7rnxn},
} -
[ss]
R. J. Sacker and G. R. Sell, "Existence of dichotomies and invariant splittings for linear differential systems. I," J. Differential Equations, vol. 15, pp. 429-458, 1974.
@ARTICLE{ss,
author = {Sacker, Robert J. and Sell, George R.},
title = {Existence of dichotomies and invariant splittings for linear differential systems. {I}},
journal = {J. Differential Equations},
fjournal = {Journal of Differential Equations},
volume = {15},
year = {1974},
pages = {429--458},
issn = {0022-0396},
mrclass = {54H20 (58F15)},
mrnumber = {0341458},
mrreviewer = {A. Morimoto},
doi = {10.1016/0022-0396(74)90067-9},
url = {https://doi.org/10.1016/0022-0396(74)90067-9},
zblnumber = {0294.58008},
} -
[ss2]
R. J. Sacker and G. R. Sell, "Existence of dichotomies and invariant splittings for linear differential systems. II," J. Differential Equations, vol. 22, iss. 2, pp. 478-496, 1976.
@ARTICLE{ss2,
author = {Sacker, Robert J. and Sell, George R.},
title = {Existence of dichotomies and invariant splittings for linear differential systems. {II}},
journal = {J. Differential Equations},
fjournal = {Journal of Differential Equations},
volume = {22},
year = {1976},
number = {2},
pages = {478--496},
issn = {0022-0396},
mrclass = {58F10 (34D35)},
mrnumber = {0440620},
mrreviewer = {A. Morimoto},
doi = {10.1016/0022-0396(76)90042-5},
url = {https://doi.org/10.1016/0022-0396(76)90042-5},
zblnumber = {0339.58013},
} -
[shenet]
W. Shen, "On the measurable dynamics of real rational functions," Ergodic Theory Dynam. Systems, vol. 23, iss. 3, pp. 957-983, 2003.
@ARTICLE{shenet,
author = {Shen, Weixiao},
title = {On the measurable dynamics of real rational functions},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {23},
year = {2003},
number = {3},
pages = {957--983},
issn = {0143-3857},
mrclass = {37F10 (37A25 37C15 37F15 37F50)},
mrnumber = {1992673},
mrreviewer = {Michael Yampolsky},
doi = {10.1017/S0143385702001311},
url = {https://doi.org/10.1017/S0143385702001311},
zblnumber = {1059.37034},
} -
[shen]
W. Shen, "Decay of geometry for unimodal maps: an elementary proof," Ann. of Math. (2), vol. 163, iss. 2, pp. 383-404, 2006.
@ARTICLE{shen,
author = {Shen, Weixiao},
title = {Decay of geometry for unimodal maps: an elementary proof},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {163},
year = {2006},
number = {2},
pages = {383--404},
issn = {0003-486X},
mrclass = {37E05 (37A99 37E20)},
mrnumber = {2199221},
mrreviewer = {Ale Jan Homburg},
doi = {10.4007/annals.2006.163.383},
url = {https://doi.org/10.4007/annals.2006.163.383},
zblnumber = {1097.37032},
} -
[sm1]
D. Smania, "Complex bounds for multimodal maps: bounded combinatorics," Nonlinearity, vol. 14, iss. 5, pp. 1311-1330, 2001.
@ARTICLE{sm1,
author = {Smania, Daniel},
title = {Complex bounds for multimodal maps: bounded combinatorics},
journal = {Nonlinearity},
fjournal = {Nonlinearity},
volume = {14},
year = {2001},
number = {5},
pages = {1311--1330},
issn = {0951-7715},
mrclass = {37E20 (37E05 37F20 37F25)},
mrnumber = {1862823},
mrreviewer = {Feliks Przytycki},
doi = {10.1088/0951-7715/14/5/320},
url = {https://doi.org/10.1088/0951-7715/14/5/320},
zblnumber = {1067.37053},
} -
[sm2]
D. Smania, "Phase space universality for multimodal maps," Bull. Braz. Math. Soc. (N.S.), vol. 36, iss. 2, pp. 225-274, 2005.
@ARTICLE{sm2,
author = {Smania, Daniel},
title = {Phase space universality for multimodal maps},
journal = {Bull. Braz. Math. Soc. (N.S.)},
fjournal = {Bulletin of the Brazilian Mathematical Society. New Series. Boletim da Sociedade Brasileira de Matem\'{a}tica},
volume = {36},
year = {2005},
number = {2},
pages = {225--274},
issn = {1678-7544},
mrclass = {37E05 (37E20)},
mrnumber = {2152018},
mrreviewer = {Fernando Fl\'{a}vio Ferreira},
doi = {10.1007/s00574-005-0038-y},
url = {https://doi.org/10.1007/s00574-005-0038-y},
zblnumber = {1100.37024},
} -
[sm3]
D. Smania, "On the hyperbolicity of the period-doubling fixed point," Trans. Amer. Math. Soc., vol. 358, iss. 4, pp. 1827-1846, 2006.
@ARTICLE{sm3,
author = {Smania, Daniel},
title = {On the hyperbolicity of the period-doubling fixed point},
journal = {Trans. Amer. Math. Soc.},
fjournal = {Transactions of the Amer. Math. Soc.},
volume = {358},
year = {2006},
number = {4},
pages = {1827--1846},
issn = {0002-9947},
mrclass = {37F25 (37E05 37E20 37F45)},
mrnumber = {2186998},
mrreviewer = {Feliks Przytycki},
doi = {10.1090/S0002-9947-05-03803-1},
url = {https://doi.org/10.1090/S0002-9947-05-03803-1},
zblnumber = {1080.37053},
} -
[sm4]
D. Smania, "Puzzle geometry and rigidity: the Fibonacci cycle is hyperbolic," J. Amer. Math. Soc., vol. 20, iss. 3, pp. 629-673, 2007.
@ARTICLE{sm4,
author = {Smania, Daniel},
title = {Puzzle geometry and rigidity: the {F}ibonacci cycle is hyperbolic},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the Amer. Math. Soc.},
volume = {20},
year = {2007},
number = {3},
pages = {629--673},
issn = {0894-0347},
mrclass = {37E20 (30C62 30C65 37C15 37F25 37F45)},
mrnumber = {2291915},
mrreviewer = {Henk Bruin},
doi = {10.1090/S0894-0347-07-00550-4},
url = {https://doi.org/10.1090/S0894-0347-07-00550-4},
zblnumber = {1222.37042},
} -
[sm5]
D. Smania, "Shy shadows of infinite-dimensional partially hyperbolic invariant sets," Ergodic Theory Dynam. Systems, vol. 39, iss. 5, pp. 1361-1400, 2019.
@ARTICLE{sm5,
author = {Smania, Daniel},
title = {Shy shadows of infinite-dimensional partially hyperbolic invariant sets},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {39},
year = {2019},
number = {5},
pages = {1361--1400},
issn = {0143-3857},
mrclass = {37C50 (37D30)},
mrnumber = {3928622},
doi = {10.1017/etds.2017.65},
url = {https://doi.org/10.1017/etds.2017.65},
zblnumber = {},
} -
[ds] D. Sullivan, "Bounds, quadratic differentials, and renormalization conjectures," in American Mathematical Society Centennial Publications, Vol. II, Amer. Math. Soc., Providence, RI, 1992, pp. 417-466.
@INCOLLECTION{ds,
author = {Sullivan, Dennis},
title = {Bounds, quadratic differentials, and renormalization conjectures},
booktitle = {American {M}athematical {S}ociety Centennial Publications, {V}ol. {II}},
venue = {{P}rovidence, {RI},
1988},
pages = {417--466},
publisher = {Amer. Math. Soc., Providence, RI},
year = {1992},
mrclass = {58F23 (30D05 39B12)},
mrnumber = {1184622},
mrreviewer = {Fuyao Ren},
zblnumber = {0936.37016},
} -
[ct] C. Tresser and P. Coullet, "Itérations d’endomorphismes et groupe de renormalisation," C. R. Acad. Sci. Paris Sér. A-B, vol. 287, iss. 7, p. a577-a580, 1978.
@ARTICLE{ct,
author = {Tresser, Charles and Coullet, Pierre},
title = {Itérations d'endomorphismes et groupe de renormalisation},
journal = {C. R. Acad. Sci. Paris Sér. A-B},
fjournal = {Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B},
volume = {287},
year = {1978},
number = {7},
pages = {A577--A580},
issn = {0151-0509},
mrclass = {58F13},
mrnumber = {0512110},
zblnumber = {0402.54046},
}