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Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication

  • CH. BONATTI (a1), S. CROVISIER (a2), L. J. DÍAZ (a3) and N. GOURMELON (a4)


Conditions are provided under which lack of domination of a homoclinic class yields robust heterodimensional cycles. Moreover, so-called viral homoclinic classes are studied. Viral classes have the property of generating copies of themselves producing wild dynamics (systems with infinitely many homoclinic classes with some persistence). Such wild dynamics also exhibits uncountably many aperiodic chain recurrence classes. A scenario (related with non-dominated dynamics) is presented where viral homoclinic classes occur. A key ingredient are adapted perturbations of a diffeomorphism along a periodic orbit. Such perturbations preserve certain homoclinic relations and prescribed dynamical properties of a homoclinic class.



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[1]Abdenur, F.. Generic robustness of spectral decompositions. Ann. Sci. Éc. Norm. Supér. 36 (2003), 213224.
[2]Abdenur, F., Bonatti, Ch. and Crovisier, S.. Nonuniform hyperbolicity for $C^1$-generic diffeomorphisms. Israel J. Math. 183 (2011), 120.
[3]Abdenur, F., Bonatti, Ch., Crovisier, S., Díaz, L. J. and Wen, L.. Periodic points and homoclinic classes. Ergod. Th. & Dynam. Sys. 27 (2007), 122.
[4]Abraham, R. and Smale, S.. Nongenericity of $\Omega $-stability. Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, CA, 1968). American Mathematical Society, Providence, RI, 1970, pp. 58.
[5]Asaoka, M.. Hyperbolic sets exhibiting $C\sp 1$-persistent homoclinic tangency for higher dimensions. Proc. Amer. Math. Soc. 136 (2008), 677686.
[6]Bochi, J. and Bonatti, Ch.. Perturbation of the Lyapunov spectra of periodic orbits. Proc. Lond. Math. Soc. (3), to appear, arXiv:1004.5029.
[7]Bonatti, Ch.. Towards a global view of dynamical systems, for the $C^1$-topology. Ergod. Th. & Dynam. Sys. 31 (2011), 959993.
[8]Bonatti, Ch. and Crovisier, S.. Récurrence et généricité. Invent. Math. 158 (2004), 33104.
[9]Bonatti, Ch. and Díaz, L. J.. Persistence of transitive diffeomorphisms. Ann. Math. 143 (1995), 367396.
[10]Bonatti, Ch. and Díaz, L. J.. On maximal transitive sets of generic diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 96 (2002), 171197.
[11]Bonatti, Ch. and Díaz, L. J.. Robust heterodimensional cycles and $C^1$-generic dynamics. J. Inst. Math. Jussieu 7 (2008), 469525.
[12]Bonatti, Ch. and Díaz, L. J.. Abundance of $C^1$-robust homoclinic tangencies. Trans. Amer. Math. Soc. to appear, arXiv:0909.4062.
[13]Bonatti, Ch., Díaz, L. J. and Kiriki, S.. Robust heterodimensional cycles and hyperbolic continuations. Nonlinearity 25 (2012), 931969.
[14]Bonatti, Ch., Díaz, L. J. and Pujals, E. R.. A ${\mathcal C}^1$-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math. (2) 158 (2003), 355418.
[15]Bonatti, Ch., Díaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity (Encyclopaedia of Mathematical Sciences: Mathematical Physics III, 102). Springer, 2004.
[16]Bonatti, Ch., Gourmelon, N. and Vivier, T.. Perturbations of the derivative along periodic orbits. Ergod. Th. & Dynam. Sys. 26 (2006), 13071337.
[17]Colli, E.. Infinitely many coexisting strange attractors. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 539579.
[18]Crovisier, S.. Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems. Ann. Math. 172 (2010), 16411667.
[19]Crovisier, S. and Pujals, E. R.. Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms, arXiv:1011.3836.
[20]Díaz, L. J., Nogueira, A. and Pujals, E. R.. Heterodimensional tangencies. Nonlinearity 19 (2006), 25432566.
[21]Franks, J.. Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc. 158 (1971), 301308.
[22]Gan, S. and Wen, L.. Heteroclinic cycles and homoclinic closures for generic diffeomorphisms. J. Dynam. Differential Equations 15 (2003), 451471.
[23]Gourmelon, N.. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete Contin. Dyn. Syst. 26 (2010), 142.
[24]Gourmelon, N.. A Franks’ lemma that preserves invariant manifolds, arXiv:0912.1121v2.
[25]Mañé, R.. Contributions to the stability conjecture. Topology 17 (1978), 383396.
[26]Moreira, C. G.. There are no $C^1$-stable intersections of regular Cantor sets. Acta Math. 206 (2011), 311323.
[27]Newhouse, S.. Diffeomorphisms with infinitely many sinks. Topology 13 (1974), 918.
[28]Newhouse, S.. The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 101151.
[29]Newhouse, S.. New phenomena associated with homoclinic tangencies. Ergod. Th. & Dynam. Sys. 24 (2004), 17251738.
[30]Pacifico, M. J., Pujals, E. R. and Vietez, J. L.. Robustly expansive homoclinic classes. Ergod. Th. & Dynam. Sys. 25 (2005), 271300.
[31]Palis, J.. A global view of dynamics and a conjecture on the denseness of finitude of attractors. Astérisque 261 (2000), 335347.
[32]Palis, J. and Takens, F.. Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and Infinitely Many Attractors (Cambridge Studies in Advanced Mathematics, 35). Cambridge University Press, Cambridge, 1993.
[33]Palis, J. and Viana, M.. High dimension diffeomorphisms displaying infinitely many periodic attractors. Ann. of Math. (2) 140 (1994), 207250.
[34]Pujals, E. R. and Sambarino, M.. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. 151 (2000), 9611023.
[35]Romero, N.. Persistence of homoclinic tangencies in higher dimensions. Ergod. Th. & Dynam. Sys. 15 (1995), 735757.
[36]Shinohara, K.. On the index problem of $C^1$-generic wild homoclinic classes in dimension three. Discrete Contin. Dyn. Syst. 31(3) (2011), 913940.
[37]Simon, C. P.. Instability in $\mathrm {Diff}(T^3)$ and the nongenericity of rational zeta functions. Trans. Amer. Math. Soc. 174 (1972), 217242.
[38]Wen, L.. Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles. Bull. Braz. Math. Soc. (N.S.) 35 (2004), 419452.


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