Abundance of 𝐶¹-robust homoclinic tangencies

Bonatti, Christian; Díaz, Lorenzo

doi:10.1090/S0002-9947-2012-05445-6

Abundance of $C^1$-robust homoclinic tangencies
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by Christian Bonatti and Lorenzo J. Díaz PDF

Trans. Amer. Math. Soc. 364 (2012), 5111-5148 Request permission

Abstract:

A diffeomorphism $f$ has a $C^1$-robust homoclinic tangency if there is a $C^1$-neighborhood $\mathcal {U}$ of $f$ such that every diffeomorphism in $g\in \mathcal {U}$ has a hyperbolic set $\Lambda _g$, depending continuously on $g$, such that the stable and unstable manifolds of $\Lambda _g$ have some non-transverse intersection. For every manifold of dimension greater than or equal to three we exhibit a local mechanism (blender-horseshoes) generating diffeomorphisms with $C^1$-robust homoclinic tangencies.

Using blender-horseshoes, we prove that homoclinic classes of $C^1$-generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings (of appropriate dimensions) display $C^1$-robust homoclinic tangencies.

References

Flavio Abdenur, Christian Bonatti, Sylvain Crovisier, and Lorenzo J. Díaz, Generic diffeomorphisms on compact surfaces, Fund. Math. 187 (2005), no. 2, 127–159. MR 2214876, DOI 10.4064/fm187-2-3
F. Abdenur, Ch. Bonatti, S. Crovisier, L. J. Díaz, and L. Wen, Periodic points and homoclinic classes, Ergodic Theory Dynam. Systems 27 (2007), no. 1, 1–22. MR 2297084, DOI 10.1017/S0143385706000538
R. Abraham and S. Smale, Nongenericity of $\Omega$-stability, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 5–8. MR 0271986
Masayuki Asaoka, Hyperbolic sets exhibiting $C^1$-persistent homoclinic tangency for higher dimensions, Proc. Amer. Math. Soc. 136 (2008), no. 2, 677–686. MR 2358509, DOI 10.1090/S0002-9939-07-09115-0
Michael Benedicks and Lennart Carleson, The dynamics of the Hénon map, Ann. of Math. (2) 133 (1991), no. 1, 73–169. MR 1087346, DOI 10.2307/2944326
Ch. Bonatti, The global dynamics of C1-generic diffeomorphisms or flows, conference in Second Latin American Congress of Mathematicians, Cancún, México (2004).
Christian Bonatti and Sylvain Crovisier, Récurrence et généricité, Invent. Math. 158 (2004), no. 1, 33–104 (French, with English and French summaries). MR 2090361, DOI 10.1007/s00222-004-0368-1
Christian Bonatti and Lorenzo J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math. (2) 143 (1996), no. 2, 357–396. MR 1381990, DOI 10.2307/2118647
Christian Bonatti and Lorenzo Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics, J. Inst. Math. Jussieu 7 (2008), no. 3, 469–525. MR 2427422, DOI 10.1017/S1474748008000030
C. Bonatti, L. J. Díaz, and E. R. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2) 158 (2003), no. 2, 355–418 (English, with English and French summaries). MR 2018925, DOI 10.4007/annals.2003.158.355
Christian Bonatti, Lorenzo J. Díaz, Enrique R. Pujals, and Jorge Rocha, Robustly transitive sets and heterodimensional cycles, Astérisque 286 (2003), xix, 187–222 (English, with English and French summaries). Geometric methods in dynamics. I. MR 2052302
Christian Bonatti, Lorenzo Justiniano Díaz, and Marcelo Viana, Discontinuity of the Hausdorff dimension of hyperbolic sets, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 6, 713–718 (English, with English and French summaries). MR 1323944
Christian Bonatti, Lorenzo J. Díaz, and Marcelo Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences, vol. 102, Springer-Verlag, Berlin, 2005. A global geometric and probabilistic perspective; Mathematical Physics, III. MR 2105774
Christian Bonatti and Marcelo Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math. 115 (2000), 157–193. MR 1749677, DOI 10.1007/BF02810585
Tomasz Downarowicz and Sheldon Newhouse, Symbolic extensions and smooth dynamical systems, Invent. Math. 160 (2005), no. 3, 453–499. MR 2178700, DOI 10.1007/s00222-004-0413-0
Nikolaz Gourmelon, Generation of homoclinic tangencies by $C^1$-perturbations, Discrete Contin. Dyn. Syst. 26 (2010), no. 1, 1–42. MR 2552776, DOI 10.3934/dcds.2010.26.1
C. G. Moreira, There are no $C^1$-stable intersections of regular Cantor sets, Acta Mathematica 206 (2011), 311–323.
Shuhei Hayashi, Connecting invariant manifolds and the solution of the $C^1$ stability and $\Omega$-stability conjectures for flows, Ann. of Math. (2) 145 (1997), no. 1, 81–137. MR 1432037, DOI 10.2307/2951824
Vadim Yu. Kaloshin, An extension of the Artin-Mazur theorem, Ann. of Math. (2) 150 (1999), no. 2, 729–741. MR 1726706, DOI 10.2307/121093
Leonardo Mora and Marcelo Viana, Abundance of strange attractors, Acta Math. 171 (1993), no. 1, 1–71. MR 1237897, DOI 10.1007/BF02392766
Sheldon E. Newhouse, Nondensity of axiom $\textrm {A}(\textrm {a})$ on $S^{2}$, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 191–202. MR 0277005
Sheldon E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 101–151. MR 556584
Jacob Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Astérisque 261 (2000), xiii–xiv, 335–347 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755446
Jacob Palis and Floris Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, vol. 35, Cambridge University Press, Cambridge, 1993. Fractal dimensions and infinitely many attractors. MR 1237641
J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math. (2) 140 (1994), no. 1, 207–250. MR 1289496, DOI 10.2307/2118546
Enrique R. Pujals and Martín Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math. (2) 151 (2000), no. 3, 961–1023. MR 1779562, DOI 10.2307/121127
Neptalí Romero, Persistence of homoclinic tangencies in higher dimensions, Ergodic Theory Dynam. Systems 15 (1995), no. 4, 735–757. MR 1346398, DOI 10.1017/S0143385700008634
Carl P. Simon, Instability in $\textrm {Diff}^{r}$ $(T^{3})$ and the nongenericity of rational zeta functions, Trans. Amer. Math. Soc. 174 (1972), 217–242. MR 317356, DOI 10.1090/S0002-9947-1972-0317356-8
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
Raúl Ures, Abundance of hyperbolicity in the $C^1$ topology, Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 6, 747–760 (English, with English and French summaries). MR 1355140
Lan Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc. (N.S.) 35 (2004), no. 3, 419–452. MR 2106314, DOI 10.1007/s00574-004-0023-x
Lan Wen, Homoclinic tangencies and dominated splittings, Nonlinearity 15 (2002), no. 5, 1445–1469. MR 1925423, DOI 10.1088/0951-7715/15/5/306
J. Yang, Ergodic measures far away from tangencies, thesis IMPA (2009).