Hyperbolic sets exhibiting 𝐶¹-persistent homoclinic tangency for higher dimensions

Asaoka, Masayuki

doi:10.1090/S0002-9939-07-09115-0

Hyperbolic sets exhibiting $C^1$-persistent homoclinic tangency for higher dimensions
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by Masayuki Asaoka

Proc. Amer. Math. Soc. 136 (2008), 677-686

DOI: https://doi.org/10.1090/S0002-9939-07-09115-0

Published electronically: October 18, 2007

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Erratum: Proc. Amer. Math. Soc. 138 (2010), 1533-1533.

Abstract:

For any manifold of dimension at least three, we give a simple construction of a hyperbolic invariant set that exhibits $C^1$-persistent homoclinic tangency. It provides an open subset of the space of $C^1$-diffeomorphisms in which generic diffeomorphisms have arbitrary given growth of the number of attracting periodic orbits and admit no symbolic extensions.

References

R. Abraham and S. Smale, Nongenericity of $\Omega$-stability, Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 5–8. MR 0271986
Christian Bonatti and Lorenzo Díaz, Connexions hétéroclines et généricité d’une infinité de puits et de sources, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 1, 135–150 (French, with English and French summaries). MR 1670524, DOI 10.1016/S0012-9593(99)80012-3
Christian Bonatti and Lorenzo Díaz, On maximal transitive sets of generic diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. 96 (2002), 171–197 (2003). MR 1985032, DOI 10.1007/s10240-003-0008-0
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
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
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
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
Vadim Yu. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys. 211 (2000), no. 1, 253–271. MR 1757015, DOI 10.1007/s002200050811
Sheldon E. Newhouse, Nondensity of axiom $\textrm {A}(\textrm {a})$ on $S^{2}$, Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 191–202. MR 0277005
Sheldon E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9–18. MR 339291, DOI 10.1016/0040-9383(74)90034-2
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 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
Clark Robinson, Dynamical systems, 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1999. Stability, symbolic dynamics, and chaos. MR 1792240