ON THE <i>F</i>-EXPANDING OF HOMOCLINIC CLASSES

Wanlou Wu; Bo Li

doi:10.11948/2156-907X.20180287

2019 Volume 9 Issue 3

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Wanlou Wu, Bo Li. ON THE F-EXPANDING OF HOMOCLINIC CLASSES[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 1083-1101. doi: 10.11948/2156-907X.20180287

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Wanlou Wu, Bo Li. ON THE F-EXPANDING OF HOMOCLINIC CLASSES[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 1083-1101. doi: 10.11948/2156-907X.20180287

ON THE F-EXPANDING OF HOMOCLINIC CLASSES

Wanlou Wu^1, ,,
Bo Li^1, ,

1.
School of Mathematical Sciences, Soochow University, No. 1 Shizijie, 215006 Suzhou, China

Corresponding authors: Email address: wuwanlou@163.com (W. Wu); Email address: libo15962221581@163.com (B. Li)

Abstract

Abstract

We establish a closing property for thin trapped (see Definition 1.2) homoclinic classes. Taking advantage of this property, we prove that if a homoclinic class $H(f, p)$ admits a dominated splitting $T_{H(f, p)}M=E\oplus_{ < }F$, where the subbundle $E$ is thin trapped with dim$E=$Ind$(p)$ and all periodic points homoclinically related to $p$ are uniformly $F$-expanding at the period (see Definition 1.1), then the subbundle $F$ is uniformly expanding.
- Homoclinic classes /
- dominated splitting /
- thin trapped /
- periodic points /
- uniformly expanding
MSC: 37D20, 37D30, 37C29

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References

[1]	C. Bonatti, L. Díaz and M. Viana, Dynamics beyond uniform hyperbolicity, Springer-Verlag, Berlin, 2005. Google Scholar
[2]	C. Bonatti, S. Gan and D. Yang, On the hyperbolicity of homoclinic classes, Discrete Contin. Dyn. Syst., 2009, 25(4), 1143-1162. doi: 10.3934/dcdsa CrossRef Google Scholar
[3]	C. Carballo, C. Morales and M. Pacifico, Homoclinic classes for generic C¹ vector fields, Ergodic Theory Dyn. Syst., 2003, 23(2), 403-415. Google Scholar
[4]	S. Crovisier, Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems, Ann. of Math., 2010, 172(3), 1641-1677. doi: 10.4007/annals CrossRef Google Scholar
[5]	S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations, Adv. Math., 2011, 226(1), 673-726. doi: 10.1016/j.aim.2010.07.013 CrossRef Google Scholar
[6]	S. Crovisier, Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms, Invent. Math., 2015, 201(2), 385-517. doi: 10.1007/s00222-014-0553-9 CrossRef Google Scholar
[7]	S. Crovisier, M. Sambarino and D. Yang Partial hyperbolicity and homoclinic tangencies, J. Eur. Math. Soc., 2015, 17(1), 1-49. doi: 10.4171/JEMS CrossRef Google Scholar
[8]	L. Díaz and B. Santoro, Collision, explosion and collapse of homoclinic classes, Nonlinearity, 2004, 17(3), 1001-1032. doi: 10.1088/0951-7715/17/3/013 CrossRef Google Scholar
[9]	S. Gan, The star systems $\mathcal{X}^$ and a proof of the* $C^1~\Omega$-stability conjecture for flows, J. Diff. Equations, 2000, 163(1), 1-17. doi: 504 Gateway Time-out The server didn't respond in time. CrossRef $\mathcal{X}^*$ and a proof of the $C^1~\Omega$-stability conjecture for flows" target="_blank">Google Scholar
[10]	S. Gan, A generalized shadowing lemma, Discrete Contin. Dyn. Syst., 2002, 8(3), 627-632. doi: 10.3934/dcdsa CrossRef Google Scholar
[11]	N. Gourmelon, Adapted metrics for dominated splittings, Ergodic Theory Dyn. Syst., 2007, 27(6), 1839-1849. doi: 10.1017/S0143385707000272 CrossRef Google Scholar
[12]	M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Springer-Verlag, Berlin-New York, 1977. Google Scholar
[13]	A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge University Press, Cambridge, 1995. Google Scholar
[14]	S. Liao, On the stability conjecture, Chinese Ann. Math., 1980, 1(1), 9-30. Google Scholar
[15]	S. Liao, On hyperbolicity properties of nonwandering sets of certain 3-dimensional differential systems, Acta Math. Sci. (English Ed.), 1983, 252(4), 361-368. Google Scholar
[16]	R. Mañé, An ergodic closing lemma, Ann. of Math., 1982, 116(3), 503-540. doi: 10.2307/2007021 CrossRef Google Scholar
[17]	R. Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics, Comm. Math. Phys., 1985, 100(4), 495-524. doi: 10.1007/BF01217727 CrossRef Google Scholar
[18]	S. Newhouse, Hyperbolic limit sets, Trans. Amer. Math. Soc., 1972, 167, 125-150. doi: 10.1090/S0002-9947-1972-0295388-6 CrossRef Google Scholar
[19]	J. Palis and W. de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. Google Scholar
[20]	S. Pilyugin, Shadowing in dynamical systems, Springer-Verlag, Berlin, 1706. Google Scholar
[21]	V. Pliss, On a conjecture of Smale, Differencialńye Uravnenija, 1972, 8, 268-282. Google Scholar
[22]	S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 1967, 73, 747-817. doi: 10.1090/S0002-9904-1967-11798-1 CrossRef Google Scholar
[23]	W. Sun and Y. Yang, Hyperbolic periodic points for chain hyperbolic homoclinic classes, Discrete Contin. Dyn. Syst., 2016, 36(7), 3911-3925. doi: 10.3934/dcdsa CrossRef Google Scholar
[24]	L. Wen, On the $C^1$ stability conjecture for flows, J. Diff. Equations, 1996, 129(2), 334-357. doi: 10.1006/jdeq.1996.0121 CrossRef $C^1$ stability conjecture for flows" target="_blank">Google Scholar
[25]	L. Wen, On the preperiodic set, Discrete Contin. Dyn. Syst., 2000, 6(1), 237-241. Google Scholar