# American Institute of Mathematical Sciences

July  2011, 29(3): 1191-1195. doi: 10.3934/dcds.2011.29.1191

## On spiral periodic points and saddles for surface diffeomorphisms

 1 Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970, Rio de Janeiro

Received  February 2010 Revised  June 2010 Published  November 2010

We prove that a $C^1$ generic orientation-preserving diffeomorphism of a closed orientable surface either is Axiom A without cycles or the closures of the sets of saddles and of periodic points without real eigenvalues have nonempty intersection.
Citation: C. Morales. On spiral periodic points and saddles for surface diffeomorphisms. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1191-1195. doi: 10.3934/dcds.2011.29.1191
##### References:
 [1] F. Abdenur, C. Bonatti, S. Crovisier and L. J. Diaz, Generic diffeomorphisms on compact surfaces, Fund. Math., 187 (2005), 127-159. doi: 10.4064/fm187-2-3.  Google Scholar [2] 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), 355-418. doi: 10.4007/annals.2003.158.355.  Google Scholar [3] F. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3.  Google Scholar [4] S. Gan, A necessary and sufficient condition for the existence of dominated splitting with a given index, Trends Math., 7 (2004), 143-168. Google Scholar [5] R. Mañé, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540. doi: 10.2307/2007021.  Google Scholar [6] C. Morales, Another dichotomy for surface diffeomorphisms, Proc. Amer. Math. Soc., 137 (2009), 2639-2644. doi: 10.1090/S0002-9939-09-09879-7.  Google Scholar [7] S. Newhouse, Lectures on dynamical systems, in "Dynamical Systems," Progress in Mathematics (CIME Lectures 1978), Birkhäuser, Boston, (1978), 1-114.  Google Scholar [8] E. R. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math. (2), 151 (2000), 961-1023. doi: 10.2307/121127.  Google Scholar [9] L. Wen, Homoclinic tangencies and dominated splittings, Nonlinearity, 15 (2002), 1445-1469. doi: 10.1088/0951-7715/15/5/306.  Google Scholar

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##### References:
 [1] F. Abdenur, C. Bonatti, S. Crovisier and L. J. Diaz, Generic diffeomorphisms on compact surfaces, Fund. Math., 187 (2005), 127-159. doi: 10.4064/fm187-2-3.  Google Scholar [2] 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), 355-418. doi: 10.4007/annals.2003.158.355.  Google Scholar [3] F. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3.  Google Scholar [4] S. Gan, A necessary and sufficient condition for the existence of dominated splitting with a given index, Trends Math., 7 (2004), 143-168. Google Scholar [5] R. Mañé, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540. doi: 10.2307/2007021.  Google Scholar [6] C. Morales, Another dichotomy for surface diffeomorphisms, Proc. Amer. Math. Soc., 137 (2009), 2639-2644. doi: 10.1090/S0002-9939-09-09879-7.  Google Scholar [7] S. Newhouse, Lectures on dynamical systems, in "Dynamical Systems," Progress in Mathematics (CIME Lectures 1978), Birkhäuser, Boston, (1978), 1-114.  Google Scholar [8] E. R. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math. (2), 151 (2000), 961-1023. doi: 10.2307/121127.  Google Scholar [9] L. Wen, Homoclinic tangencies and dominated splittings, Nonlinearity, 15 (2002), 1445-1469. doi: 10.1088/0951-7715/15/5/306.  Google Scholar
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