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February  2013, 33(2): 879-884. doi: 10.3934/dcds.2013.33.879

A note on a sifting-type lemma

Xiao Wen 1, and  Xiongping Dai 2,

1. 

School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
2. 

Department of Mathematics, Nanjing University, Nanjing, 210093

Received  August 2011 Revised  December 2011 Published  September 2012

In this note, we improve a combinatorial sifting-type lemma obtained in [11].More precisely, we sift out a continuous infinite "$(\xi_1,\xi_2)$-Liao string" sequence for any real sequence $\{a_i\}_1^\infty$ with $\limsup_{n\to\infty}{n}^{-1}\sum_{i=1}^na_i=\xi\in(\xi_1,\xi_2)$.
Keywords: Quasi-hyperbolic string, sifting-type lemma..
Mathematics Subject Classification: 37D3.
Citation: Xiao Wen, Xiongping Dai. A note on a sifting-type lemma. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 879-884. doi: 10.3934/dcds.2013.33.879
References:
[1]

J. Alves and V. Araújo, Hyperbolic times: frequency versus integrability, Ergod. Th. & Dynam. Sys., 24 (2004), 329-346. doi: 10.1017/S0143385703000555.  Google Scholar

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S. Gan, A generalized shadowing lemma, Discrete Cont. Dynam. Syst., 8 (2002), 627-632.  Google Scholar

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S. Gan and L. Wen, Nonsingular star flows satisfy axiom A and the no-cycle condition, Invent. Math., 164 (2006), 279-315. doi: 10.1007/s00222-005-0479-3.  Google Scholar

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S. Liao, An existence theorem for periodic orbits, (Chinese) Beijing Daxue Xuebao, 1 (1979), 1-20.  Google Scholar

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S. Liao, On the stability conjecture, Chinese Annals of Math., 1 (1980), 9-30.  Google Scholar

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V. Pliss, On a conjecture of Smale, Diff. Uravnenija, 8 (1972), 268-282.  Google Scholar

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R. Potrie and M. Sambarino, Codimension one generic homoclinic classes with interior, Bull. Braz. Math. Soc., 41 (2010), 125-138.doi: 10.1007/s00574-010-0006-z doi: 10.1007/s00574-010-0006-z.  Google Scholar

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L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc., 35 (2004), 419-452.  Google Scholar

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L. Wen, The selecting lemma of Liao, Discrete Cont. Dynam. Syst., 20 (2008), 159-175. doi: 10.3934/dcds.2008.20.159.  Google Scholar

[11]

X. Wen, S. Gan and L. Wen, $C^1$-stably shadowable chain components are hyperbolic, J. Differential Equations, 246 (2009), 340-357. doi: 10.1016/j.jde.2008.03.032.  Google Scholar

[12]

P. Zhang, A diffeomorphism with global dominated splitting can not be minimal, Proc. Amer. Math. Soc., 140 (2012), 589-593.  Google Scholar

show all references

References:
[1]

J. Alves and V. Araújo, Hyperbolic times: frequency versus integrability, Ergod. Th. & Dynam. Sys., 24 (2004), 329-346. doi: 10.1017/S0143385703000555.  Google Scholar

[2]

S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations, Advances in Mathematics, 226 (2011), 673-726. doi: 10.1016/j.aim.2010.07.013.  Google Scholar

[3]

S. Gan, A generalized shadowing lemma, Discrete Cont. Dynam. Syst., 8 (2002), 627-632.  Google Scholar

[4]

S. Gan and L. Wen, Nonsingular star flows satisfy axiom A and the no-cycle condition, Invent. Math., 164 (2006), 279-315. doi: 10.1007/s00222-005-0479-3.  Google Scholar

[5]

S. Liao, An existence theorem for periodic orbits, (Chinese) Beijing Daxue Xuebao, 1 (1979), 1-20.  Google Scholar

[6]

S. Liao, On the stability conjecture, Chinese Annals of Math., 1 (1980), 9-30.  Google Scholar

[7]

V. Pliss, On a conjecture of Smale, Diff. Uravnenija, 8 (1972), 268-282.  Google Scholar

[8]

R. Potrie and M. Sambarino, Codimension one generic homoclinic classes with interior, Bull. Braz. Math. Soc., 41 (2010), 125-138.doi: 10.1007/s00574-010-0006-z doi: 10.1007/s00574-010-0006-z.  Google Scholar

[9]

L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc., 35 (2004), 419-452.  Google Scholar

[10]

L. Wen, The selecting lemma of Liao, Discrete Cont. Dynam. Syst., 20 (2008), 159-175. doi: 10.3934/dcds.2008.20.159.  Google Scholar

[11]

X. Wen, S. Gan and L. Wen, $C^1$-stably shadowable chain components are hyperbolic, J. Differential Equations, 246 (2009), 340-357. doi: 10.1016/j.jde.2008.03.032.  Google Scholar

[12]

P. Zhang, A diffeomorphism with global dominated splitting can not be minimal, Proc. Amer. Math. Soc., 140 (2012), 589-593.  Google Scholar

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