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

A note on a sifting-type lemma

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 $\limsup_{n\to\infty}{n}^{-1}\sum_{i=1}^na_i=\xi\in(\xi_1,\xi_2)$.
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:
[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.

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.

[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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