2006, 14(3): 409-417. doi: 10.3934/dcds.2006.14.409

Necessary and sufficient conditions for semi-uniform ergodic theorems and their applications

1. 

Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080, China

2. 

School of Mathematical Sciences, Peking University, Beijing, 100871, China

3. 

School of Sciences, Beihang University, Beijing, 100083, China

Received  July 2004 Revised  September 2005 Published  December 2005

It has been established one-side uniform convergence in both the Birkhoff and sub-additive ergodic theorems under conditions on growth rates with respect to all the invariant measures. In this paper we show these conditions are both necessary and sufficient. These results are applied to study quasiperiodically forced systems. Some meaningful geometric properties of invariant sets of such systems are presented. We also show that any strange compact invariant set of a $\mathcal{C}^1$ quasiperiodically forced system must support an invariant measure with a non-negative normal Lyapunov exponent.
Citation: Zuohuan Zheng, Jing Xia, Zhiming Zheng. Necessary and sufficient conditions for semi-uniform ergodic theorems and their applications. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 409-417. doi: 10.3934/dcds.2006.14.409
[1]

Mariko Arisawa, Hitoshi Ishii. Some properties of ergodic attractors for controlled dynamical systems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 43-54. doi: 10.3934/dcds.1998.4.43

[2]

Àlex Haro. On strange attractors in a class of pinched skew products. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 605-617. doi: 10.3934/dcds.2012.32.605

[3]

Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalizable Expanding Baker Maps: Coexistence of strange attractors. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1651-1678. doi: 10.3934/dcds.2017068

[4]

Shin Kiriki, Yusuke Nishizawa, Teruhiko Soma. Heterodimensional tangencies on cycles leading to strange attractors. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 285-300. doi: 10.3934/dcds.2010.27.285

[5]

Tingting Zhang, Àngel Jorba, Jianguo Si. Weakly hyperbolic invariant tori for two dimensional quasiperiodically forced maps in a degenerate case. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6599-6622. doi: 10.3934/dcds.2016086

[6]

Tanja Eisner, Jakub Konieczny. Automatic sequences as good weights for ergodic theorems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4087-4115. doi: 10.3934/dcds.2018178

[7]

Flaviano Battelli, Michal Fe?kan. Chaos in forced impact systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 861-890. doi: 10.3934/dcdss.2013.6.861

[8]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1

[9]

Yuri Kifer. Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2687-2716. doi: 10.3934/dcds.2018113

[10]

Julia Brettschneider. On uniform convergence in ergodic theorems for a class of skew product transformations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 873-891. doi: 10.3934/dcds.2011.29.873

[11]

Hunseok Kang. Dynamics of local map of a discrete Brusselator model: eventually trapping regions and strange attractors. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 939-959. doi: 10.3934/dcds.2008.20.939

[12]

Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 941-966. doi: 10.3934/dcds.2018040

[13]

Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Persistent two-dimensional strange attractors for a two-parameter family of Expanding Baker Maps. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-14. doi: 10.3934/dcdsb.2018201

[14]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785

[15]

Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i

[16]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

[17]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017

[18]

Michal Fečkan. Bifurcation from degenerate homoclinics in periodically forced systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 359-374. doi: 10.3934/dcds.1999.5.359

[19]

Alexander Krasnosel'skii. Resonant forced oscillations in systems with periodic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 239-254. doi: 10.3934/dcds.2013.33.239

[20]

Renato C. Calleja, Alessandra Celletti, Rafael de la Llave. Construction of response functions in forced strongly dissipative systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4411-4433. doi: 10.3934/dcds.2013.33.4411

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]