A note on a sifting-type lemma

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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

Abstract
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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 and Continuous Dynamical Systems, 2013, 33 (2) : 879-884. doi: 10.3934/dcds.2013.33.879

References:

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[3]	S. Gan, A generalized shadowing lemma, Discrete Cont. Dynam. Syst., 8 (2002), 627-632.
[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.
[5]	S. Liao, An existence theorem for periodic orbits, (Chinese) Beijing Daxue Xuebao, 1 (1979), 1-20.
[6]	S. Liao, On the stability conjecture, Chinese Annals of Math., 1 (1980), 9-30.
[7]	V. Pliss, On a conjecture of Smale, Diff. Uravnenija, 8 (1972), 268-282.
[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.
[9]	L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc., 35 (2004), 419-452.
[10]	L. Wen, The selecting lemma of Liao, Discrete Cont. Dynam. Syst., 20 (2008), 159-175. doi: 10.3934/dcds.2008.20.159.
[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.
[12]	P. Zhang, A diffeomorphism with global dominated splitting can not be minimal, Proc. Amer. Math. Soc., 140 (2012), 589-593.

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References:

[1]	J. Alves and V. Araújo, Hyperbolic times: frequency versus integrability, Ergod. Th. & Dynam. Sys., 24 (2004), 329-346. doi: 10.1017/S0143385703000555.
[2]	S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations, Advances in Mathematics, 226 (2011), 673-726. doi: 10.1016/j.aim.2010.07.013.
[3]	S. Gan, A generalized shadowing lemma, Discrete Cont. Dynam. Syst., 8 (2002), 627-632.
[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.
[5]	S. Liao, An existence theorem for periodic orbits, (Chinese) Beijing Daxue Xuebao, 1 (1979), 1-20.
[6]	S. Liao, On the stability conjecture, Chinese Annals of Math., 1 (1980), 9-30.
[7]	V. Pliss, On a conjecture of Smale, Diff. Uravnenija, 8 (1972), 268-282.
[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.
[9]	L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc., 35 (2004), 419-452.
[10]	L. Wen, The selecting lemma of Liao, Discrete Cont. Dynam. Syst., 20 (2008), 159-175. doi: 10.3934/dcds.2008.20.159.
[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.
[12]	P. Zhang, A diffeomorphism with global dominated splitting can not be minimal, Proc. Amer. Math. Soc., 140 (2012), 589-593.

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