January  2008, 20(1): 159-175. doi: 10.3934/dcds.2008.20.159

The selecting Lemma of Liao

1. 

School of Mathematic Sciences, Peking University, Beijing, 100871

Received  February 2007 Revised  August 2007 Published  October 2007

The selecting lemma of Liao selects, under certain conditions of a non-hyperbolic setting, a special kind of orbits of finite length, called quasi-hyperbolic strings, which can be shadowed by true orbits. In this article we give an exposition on this lemma, and illustrate some recent applications.
Citation: Lan Wen. The selecting Lemma of Liao. Discrete and Continuous Dynamical Systems, 2008, 20 (1) : 159-175. doi: 10.3934/dcds.2008.20.159
[1]

Zhiping Li, Yunhua Zhou. Quasi-shadowing for partially hyperbolic flows. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2089-2103. doi: 10.3934/dcds.2020107

[2]

Dante Carrasco-Olivera, Bernardo San Martín. Robust attractors without dominated splitting on manifolds with boundary. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4555-4563. doi: 10.3934/dcds.2014.34.4555

[3]

Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006

[4]

Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421

[5]

Shaobo Gan, Kazuhiro Sakai, Lan Wen. $C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 205-216. doi: 10.3934/dcds.2010.27.205

[6]

Fang Zhang, Yunhua Zhou. On the limit quasi-shadowing property. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2861-2879. doi: 10.3934/dcds.2017123

[7]

Xinsheng Wang, Lin Wang, Yujun Zhu. Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2125-2140. doi: 10.3934/dcds.2018087

[8]

Pedro Duarte, Silvius Klein. Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5379-5387. doi: 10.3934/dcds.2018237

[9]

Xiao Wen, Lan Wen. No-shadowing for singular hyperbolic sets with a singularity. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 6043-6059. doi: 10.3934/dcds.2020258

[10]

Jingyu Li, Chuangchuang Liang. Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 833-849. doi: 10.3934/dcds.2016.36.833

[11]

Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089

[12]

Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365

[13]

Chengming Cao, Xiaoping Yuan. Quasi-periodic solutions for perturbed generalized nonlinear vibrating string equation with singularities. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1867-1901. doi: 10.3934/dcds.2017079

[14]

Roberto Triggiani. Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: the clamped case. Conference Publications, 2007, 2007 (Special) : 993-1004. doi: 10.3934/proc.2007.2007.993

[15]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control and Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002

[16]

Guangcun Lu. Parameterized splitting theorems and bifurcations for potential operators, Part II: Applications to quasi-linear elliptic equations and systems. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1317-1368. doi: 10.3934/dcds.2021155

[17]

Yujun Zhu. Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 869-882. doi: 10.3934/dcds.2014.34.869

[18]

Harry Crimmins. Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2795-2857. doi: 10.3934/dcds.2022001

[19]

Sergei Yu. Pilyugin. Variational shadowing. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 733-737. doi: 10.3934/dcdsb.2010.14.733

[20]

Sergei A. Avdonin, Boris P. Belinskiy. Controllability of a string under tension. Conference Publications, 2003, 2003 (Special) : 57-67. doi: 10.3934/proc.2003.2003.57

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (125)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]