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April  2013, 33(4): 1375-1388. doi: 10.3934/dcds.2013.33.1375

Observable optimal state points of subadditive potentials

Eleonora Catsigeras 1, and  Yun Zhao 2,

1. 

Instituto de Matemática y Estadística Rafael Laguardia, Universidad de la República, Av. Herrera y Reissig 565, C.P.11300, Montevideo, Uruguay
2. 

Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu

Received  October 2011 Revised  June 2012 Published  October 2012

For a sequence of subadditive potentials, a method of choosing state points with negative growth rates for an ergodic dynamical system was given in [5]. This paper first generalizes this result to the non-ergodic dynamics, and then proves that under some mild additional hypothesis, one can choose points with negative growth rates from a positive Lebesgue measure set, even if the system does not preserve any measure that is absolutely continuous with respect to Lebesgue measure.
Keywords: subadditive potentials, Optimal state points, observable measures., milnor-like attractors.
Mathematics Subject Classification: 37A30, 37L40, 37A05, 37K4.
Citation: Eleonora Catsigeras, Yun Zhao. Observable optimal state points of subadditive potentials. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1375-1388. doi: 10.3934/dcds.2013.33.1375
References:
[1]

Y. Cao, On growth rates of sub-additive functions for semi-flows: determined and random cases,, J. Diff. Eqns, 231 (2006), 1.  doi: 10.1016/j.jde.2006.08.016.  Google Scholar

[2]

E. Catsigeras and H. Enrich, SRB-like measures for $C^0$ dynamics,, Bull. Polish Acad. Sci. Math., 59 (2011), 151.  doi: 10.4064/ba59-2-5.  Google Scholar

[3]

E. Catsigeras, Milnor-like attractors,, preprint, ().   Google Scholar

[4]

X. Dai, Y. Huang and M. Xiao, Periodically switched stability induces exponential stability of discrete-time linear switched systems in the sense of Markovian probabilities,, Automatica, 47 (2011), 1512.  doi: 10.1016/j.automatica.2011.02.034.  Google Scholar

[5]

X. Dai, Optimal state points of the subadditive ergodic theorem,, Nonlinearity, 24 (2011), 1565.  doi: 10.1088/0951-7715/24/5/009.  Google Scholar

[6]

T. Golenishcheva-Kutuzova and V. Kleptsyn, Convergence of the Krylov-Bogolyubov procedure in Bowan's example,, (Russian) Mat. Zametki, 82 (2007), 678.   Google Scholar

[7]

A. Katok and B. Hasselblatt, "An Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and Its Applications,", volume 54, (1995).   Google Scholar

[8]

J. F. C. Kingman, Subadditive ergodic theory,, Ann. Probab., 1 (1973), 883.   Google Scholar

[9]

M. Misiurewicz, Ergodic natural measures,, in, (2005), 1.   Google Scholar

[10]

S. J. Schreiber, On growth rates of subadditive functions for semi-flows,, J. Diff. Eqns, 148 (1998), 334.  doi: 10.1006/jdeq.1998.3471.  Google Scholar

[11]

K. Sigmund, Generic properties of invariant measures for axiom A-diffeomorphisms,, Inventiones Math., 11 (1970), 99.  doi: 10.1007/BF01404606.  Google Scholar

[12]

K. Sigmund, On the distribution of periodic points for $\beta-$shifts,, Monatsh. Math, 82 (1976), 247.  doi: 10.1007/BF01526329.  Google Scholar

[13]

R. Sturman, and J. Stark, Semi-uniform ergodic theorems and applications to forced systems,, Nonlinearity, 13 (2000), 113.   Google Scholar

[14]

Y. Takahashi, Entropy Functional (free energy) for Dynamical Systems and their Random Perturbations,, in, 32 (1982), 437.   Google Scholar

[15]

F. Takens, Heteroclinic attractors: time averages and moduli of topological conjugacy,, Bol. Soc. Brasil. Math., 25 (1994), 107.  doi: 10.1007/BF01232938.  Google Scholar

[16]

P. Walters, "An Introduction to Ergodic Theory,", GTM 79, (1982).   Google Scholar

show all references

References:
[1]

Y. Cao, On growth rates of sub-additive functions for semi-flows: determined and random cases,, J. Diff. Eqns, 231 (2006), 1.  doi: 10.1016/j.jde.2006.08.016.  Google Scholar

[2]

E. Catsigeras and H. Enrich, SRB-like measures for $C^0$ dynamics,, Bull. Polish Acad. Sci. Math., 59 (2011), 151.  doi: 10.4064/ba59-2-5.  Google Scholar

[3]

E. Catsigeras, Milnor-like attractors,, preprint, ().   Google Scholar

[4]

X. Dai, Y. Huang and M. Xiao, Periodically switched stability induces exponential stability of discrete-time linear switched systems in the sense of Markovian probabilities,, Automatica, 47 (2011), 1512.  doi: 10.1016/j.automatica.2011.02.034.  Google Scholar

[5]

X. Dai, Optimal state points of the subadditive ergodic theorem,, Nonlinearity, 24 (2011), 1565.  doi: 10.1088/0951-7715/24/5/009.  Google Scholar

[6]

T. Golenishcheva-Kutuzova and V. Kleptsyn, Convergence of the Krylov-Bogolyubov procedure in Bowan's example,, (Russian) Mat. Zametki, 82 (2007), 678.   Google Scholar

[7]

A. Katok and B. Hasselblatt, "An Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and Its Applications,", volume 54, (1995).   Google Scholar

[8]

J. F. C. Kingman, Subadditive ergodic theory,, Ann. Probab., 1 (1973), 883.   Google Scholar

[9]

M. Misiurewicz, Ergodic natural measures,, in, (2005), 1.   Google Scholar

[10]

S. J. Schreiber, On growth rates of subadditive functions for semi-flows,, J. Diff. Eqns, 148 (1998), 334.  doi: 10.1006/jdeq.1998.3471.  Google Scholar

[11]

K. Sigmund, Generic properties of invariant measures for axiom A-diffeomorphisms,, Inventiones Math., 11 (1970), 99.  doi: 10.1007/BF01404606.  Google Scholar

[12]

K. Sigmund, On the distribution of periodic points for $\beta-$shifts,, Monatsh. Math, 82 (1976), 247.  doi: 10.1007/BF01526329.  Google Scholar

[13]

R. Sturman, and J. Stark, Semi-uniform ergodic theorems and applications to forced systems,, Nonlinearity, 13 (2000), 113.   Google Scholar

[14]

Y. Takahashi, Entropy Functional (free energy) for Dynamical Systems and their Random Perturbations,, in, 32 (1982), 437.   Google Scholar

[15]

F. Takens, Heteroclinic attractors: time averages and moduli of topological conjugacy,, Bol. Soc. Brasil. Math., 25 (1994), 107.  doi: 10.1007/BF01232938.  Google Scholar

[16]

P. Walters, "An Introduction to Ergodic Theory,", GTM 79, (1982).   Google Scholar

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