Nonexistence of wandering domains for strongly dissipative infinitely renormalizable Hénon maps at the boundary of chaos

Dyi-Shing Ou1
1Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794-3651, USA

Tóm tắt

Từ khóa


Tài liệu tham khảo

Arosio, L., Benini, A.M., Fornæss, J.E., Peters, H.: Dynamics of transcendental Hénon maps. Math. Ann. (2018). https://doi.org/10.1007/s00208-018-1643-6

Astorg, M., Buff, X., Dujardin, R., Peters, H., Raissy, J.: A two-dimensional polynomial mapping with a wandering Fatou component. Ann. Math. 184, 263–313 (2016). https://doi.org/10.4007/annals.2016.184.1.2

Baker, I.N.: An entire function which has wandering domains. J. Aust. Math. Soc. A 22(02), 173–176 (1976). https://doi.org/10.1017/S1446788700015287

Bedford, E.: Dynamics of polynomial diffeomorphisms of $${\mathbb{C}}^{2}$$: Foliations and laminations. arXiv:1501.01402 (2015)

Blokh, A.M., Lyubich, M.Y.: Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems 2. Ergod. Theor. Dyn. Syst. 9(04), 751–758 (1989). https://doi.org/10.1017/S0143385700005319

Bonatti, C., Gambaudo, J.M., Lion, J.M., Tresser, C.: Wandering domains for infinitely renormalizable diffeomorphisms of the disk. Proc. Am. Math. Soc. (1994). https://doi.org/10.1090/S0002-9939-1994-1223264-0

Brin, M., Stuck, G.: Introduction to Dynamical Systems. Cambridge University Press, Cambridge (2002). https://doi.org/10.1017/CBO9780511755316

Campanino, M., Epstein, H., Ruelle, D.: On Feigenbaum’s functional equation $$g \circ g (\lambda x)+ \lambda g (x)= 0$$. Topology 21(2), 125–129 (1982). https://doi.org/10.1016/0040-9383(82)90001-5

Colli, E., Vargas, E.: Non-trivial wandering domains and homoclinic bifurcations. Ergod. Theory Dyn. Syst. 21(06), 1657–1681 (2001). https://doi.org/10.1017/S0143385701001791

Cremer, H.: Über die schrödersche funktionalgleichung und das schwarzsche eckenabbildungsproblem. Leipziger Berichte 84, 291–324 (1932)

de Carvalho, A., Lyubich, M., Martens, M.: Renormalization in the Hénon family, I: universality but non-rigidity. J. Stat. Phys. 121(5–6), 611–669 (2005). https://doi.org/10.1007/s10955-005-8668-4

de Faria, E., de Melo, W., Pinto, A.: Global hyperbolicity of renormalization for $${C}^r$$ unimodal mappings. Ann. Math. (2006). https://doi.org/10.4007/annals.2006.164.731

de Melo, W.: A finiteness problem for one-dimensional maps. Proc. Am. Math. Soc. 101(4), 721–727 (1987). https://doi.org/10.1090/S0002-9939-1987-0911040-1

de Melo, W., Pinto, A.A.: Rigidity of $${C}^2$$ infinitely renormalizable unimodal maps. Commun. Math. Phys. 208(1), 91–105 (1999). https://doi.org/10.1007/s002200050749

de Melo, W., van Strien, S.: One-dimensional dynamics: the Schwarzian derivative and beyond. Bull. Am. Math. Soc. 18(2), 159–162 (1988). https://doi.org/10.1090/S0273-0979-1988-15633-9

de Melo, W., van Strien, S.: A structure theorem in one dimensional dynamics. Ann. Math. 129(3), 519–546 (1989). https://doi.org/10.2307/1971516

de Melo, W., van Strien, S.: One-Dimensional Dynamics. Springer, Berlin (1993). https://doi.org/10.1007/978-3-642-78043-1

Denjoy, A.: Sur les courbes définies par les équations différentielles à la surface du tore. Journal de Mathématiques Pures et Appliquées 11, 333–376 (1932)

Dujardin, R.: Hénon-like mappings in $${\mathbb{C}}^{2}$$. Am. J. Math. 126(2), 439–472 (2004). https://doi.org/10.1353/ajm.2004.0010

Epstein, H., Lascoux, J.: Analyticity properties of the Feigenbaum function. Commun. Math. Phys. 81(3), 437–453 (1981). https://doi.org/10.1007/BF01209078

Eremenko, A.E., Lyubich, M.Y.: Examples of entire functions with pathological dynamics. J. Lond. Math. Soc. (2) 36(3), 458–468 (1987). https://doi.org/10.1112/jlms/s2-36.3.458

Eremenko, A.E., Lyubich, M.Y.: Dynamical properties of some classes of entire functions. Annales de l’institut Fourier 42(4), 989–1020 (1992). https://doi.org/10.5802/aif.1318

Fatou, P.: Sur les équations fonctionnelles. Soc. Math. France Bull. 47, 161–271 (1919)

Fatou, P.: Sur les équations fonctionnelles. Soc. Math. France Bull. 48, 33–94 (1920)

Fatou, P.: Sur les équations fonctionnelles. Soc. Math. France Bull. 48, 208–314 (1920)

Fornæss, J.E., Sibony, N.: Fatou and Julia sets for entire mappings in $${\mathbb{C}}^{k}$$. Math. Ann. 311(1), 27–40 (1998). https://doi.org/10.1007/s002080050174

Friedland, S., Milnor, J.: Dynamical properties of plane polynomial automorphisms. Ergod. Theory Dyn. Syst. 9(1), 67–99 (1989). https://doi.org/10.1017/S014338570000482X

Gambaudo, J.M., van Strien, S., Tresser, C.: Hénon-like maps with strange attractors: there exist $${C}^{\infty }$$ Kupka-Smale diffeomorphisms on $${S}^2$$ with neither sinks nor sources. Nonlinearity 2(2), 287 (1989). https://doi.org/10.1088/0951-7715/2/2/005

Goldberg, L.R., Keen, L.: A finiteness theorem for a dynamical class of entire functions. Ergod. Theory Dyn. Syst. 6(02), 183–192 (1986). https://doi.org/10.1017/S0143385700003394

Guckenheimer, J.: Sensitive dependence to initial conditions for one dimensional maps. Commun. Math. Phys. 70(2), 133–160 (1979). https://doi.org/10.1007/BF01982351

Hall, G.R.: A $${C}^{\infty }$$ Denjoy counterexample. Ergod. Theory Dyn. Syst. 1(03), 261–272 (1981). https://doi.org/10.1017/S0143385700001243

Harrison, J.: Unsmoothable diffeomorphisms. Ann. Math. 102(1), 85–94 (1975). https://doi.org/10.2307/1970975

Harrison, J.: Unsmoothable diffeomorphisms on higher dimensional manifolds. Proc. Am. Math. Soc. 73(2), 249–255 (1979). https://doi.org/10.1090/S0002-9939-1979-0516473-9

Hazard, P.: Hénon-like maps with arbitrary stationary combinatorics. Ergod. Theory Dyn. Syst. 31(05), 1391–1443 (2011). https://doi.org/10.1017/S0143385710000398

Hazard, P., Martens, M., Tresser, C.: Infinitely many moduli of stability at the dissipative boundary of chaos. Trans. Am. Math. Soc. 370(1), 27–51 (2018). https://doi.org/10.1090/tran/6940

Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50(1), 69–77 (1976). https://doi.org/10.1007/BF01608556

Herman, M.R.: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publications Mathématiques de l’Institut des Hautes Études Scientifiques 49(1), 5–233 (1979). https://doi.org/10.1007/BF02684798

Hu, J., Sullivan, D.P.: Topological conjugacy of circle diffeomorphisms. Ergod. Theory Dyn. Syst. 17(01), 173–186 (1997). https://doi.org/10.1017/s0143385797061002

Hubbard, J.H., Oberste-Vorth, R.W.: Hénon mappings in the complex domain. In: Real and Complex Dynamical Systems, pp. 89–132. Springer (1995). https://doi.org/10.1007/978-94-015-8439-5_5

Hubbard, J.H.: The Hénon mapping in the complex domain. In: Barnsley, M.F., Demko, S.G. (eds.) Chaotic Dynamics and Fractals, pp. 101–111. Academic Press, New York (1986). https://doi.org/10.1016/B978-0-12-079060-9.50010-2

Kiriki, S., Soma, T.: Existence of generic cubic homoclinic tangencies for Hénon maps. Ergod. Theory Dyn. Syst. 33(4), 1029–1051 (2013). https://doi.org/10.1017/S0143385712000168

Kiriki, S., Soma, T.: Takens’ last problem and existence of non-trivial wandering domains. Adv. Math. 306, 524–588 (2017). https://doi.org/10.1016/j.aim.2016.10.019

Kiriki, S., Li, M.C., Soma, T.: Coexistence of invariant sets with and without SRB measures in Hénon family. Nonlinearity 23(9), 2253 (2010). https://doi.org/10.1088/0951-7715/23/9/010

Kiriki, S., Nakano, Y., Soma, T.: Non-trivial wandering domains for heterodimensional cycles. Nonlinearity 30(8), 3255 (2017). https://doi.org/10.1088/1361-6544/aa7cc6

Kwakkel, F.H.: Surface homeomorphisms: the interplay between topology, geometry and dynamics. Phd, University of Warwick (2009). http://webcat.warwick.ac.uk/record=b2317854~S15

Kwakkel, F., Markovic, V.: Topological entropy and diffeomorphisms of surfaces with wandering domains. Ann. Acad. Sci. Fenn. Math. 35(2), 503–513 (2010). https://doi.org/10.5186/aasfm.2010.3531

Lyubich, M.: Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems: 1. The case of negative Schwarzian derivative. Ergod. Theory Dyn. Syst. 9(04), 737–749 (1989). https://doi.org/10.1017/S0143385700005307

Lyubich, M., Martens, M.: Renormalization in the Hénon family, II: the heteroclinic web. Invent. Math. 186(1), 115–189 (2011). https://doi.org/10.1007/s00222-011-0316-9

MacKay, R.S., Tresser, C.: Boundary of topological chaos for bimodal maps of the interval. J. Lond. Math. Soc. 2(1), 164–181 (1988). https://doi.org/10.1112/jlms/s2-37.121.164

Martens, M., Winckler, B.: Instability of renormalization. arXiv:1609.04473 (2016)

Martens, M., Winckler, B.: On the hyperbolicity of Lorenz renormalization. Commun. Math. Phys. 325(1), 185–257 (2014). https://doi.org/10.1007/s00220-013-1858-z

Martens, M., de Melo, W., van Strien, S.: Julia-Fatou-Sullivan theory for real one-dimensional dynamics. Acta Math. 168(1), 273–318 (1992). https://doi.org/10.1007/BF02392981

McSwiggen, P.D.: Diffeomorphisms of the torus with wandering domains. Proc. Am. Math. Soc. 117(4), 1175–1186 (1993). https://doi.org/10.1090/S0002-9939-1993-1154247-6

McSwiggen, P.D.: Diffeomorphisms of the k-torus with wandering domains. Ergod. Theory Dyn. Syst. 15(06), 1189–1205 (1995). https://doi.org/10.1017/S014338570000986X

Milnor, J., Thurston, W.: On iterated maps of the interval. In: Dynamical systems, pp. 465–563. Springer (1988). https://doi.org/10.1007/BFb0082847

Milnor, J.: Dynamics in One Complex Variable, 3rd edn. Princeton University Press, Princeton (2011). https://doi.org/10.2307/j.ctt7rnxn

Navas, A.: Wandering domains for diffeomorphisms of the k-torus: a remark on a theorem by Norton and Sullivan. arXiv:1702.02251 (2017)

Norton, A.: An area approach to wandering domains for smooth surface endomorphisms. Ergod. Theory Dyn. Syst. 11(01), 181–187 (1991). https://doi.org/10.1017/S0143385700006064

Norton, A.: Denjoy’s theorem with exponents. Proc. Am. Math. Soc. 127(10), 3111–3118 (1999). https://doi.org/10.1090/S0002-9939-99-04852-2

Norton, A., Sullivan, D.: Wandering domains and invariant conformal structures for mappings of the 2-torus. Ann. Acad. Sci. Fenn. Math 21(1), 51–68 (1996)

Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle (i,ii,iii,iv). Journal de Mathématiques Pures et Appliquées (1881,82,85,86)

Raissy, J.: Polynomial skew-products in dimension 2: bulging and wandering Fatou components. Bollettino dell’Unione Matematica Italiana (2016). https://doi.org/10.1007/s40574-016-0101-1

Schwartz, A.J.: A generalization of a Poincaré–Bendixson theorem to closed two-dimensional manifolds. Am. J. Math. (1963). https://doi.org/10.2307/2373135

Sharkovskii, A.N., Ivanov, A.F.: $${C}^{\infty }$$-mappings of an interval with attracting cycles of arbitrarily large periods. Ukr. Math. J. 35(4), 455–458 (1983). https://doi.org/10.1007/BF01093104

Sullivan, D.: Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou-Julia problem on wandering domains. Ann. Math. 122(2), 401–418 (1985). https://doi.org/10.2307/1971308

van Strien, S.: One-dimensional dynamics in the new millennium. Discrete Contin. Dyn. Syst. 27(2), 557–588 (2010). https://doi.org/10.3934/dcds.2010.27.557

Yoccoz, J.C.: Sur la disparition de la propriété de Denjoy-Koksma en dimension 2. C. R. Acad. Sci. Paris Sér. A-B 291, A655–A658 (1980)

Yoccoz, J.C.: Il n’ya pas de contre-exemple de Denjoy analytique. C. R. Acad. Sci. Paris Sér. I Math. 298(7), 141–144 (1984)