# American Institute of Mathematical Sciences

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

## Observable optimal state points of subadditive potentials

 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.
Citation: Eleonora Catsigeras, Yun Zhao. Observable optimal state points of subadditive potentials. Discrete & Continuous Dynamical Systems, 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), 1-17. 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), 151-164. 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), 1512-1519. doi: 10.1016/j.automatica.2011.02.034.  Google Scholar [5] X. Dai, Optimal state points of the subadditive ergodic theorem, Nonlinearity, 24 (2011), 1565-1573. 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), 678-689; Translation in Math. Notes, 82 (2007), 608-618.  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, Cambridge: Cambridge University Press, 1995.  Google Scholar [8] J. F. C. Kingman, Subadditive ergodic theory, Ann. Probab., 1 (1973), 883-909.  Google Scholar [9] M. Misiurewicz, Ergodic natural measures, in "Algebraic and Topological Dynamics", volume 385, of Contemp. Math., 1-6 Amer. Math. Soc. Providence R. I. 2005.  Google Scholar [10] S. J. Schreiber, On growth rates of subadditive functions for semi-flows, J. Diff. Eqns, 148 (1998), 334-350. doi: 10.1006/jdeq.1998.3471.  Google Scholar [11] K. Sigmund, Generic properties of invariant measures for axiom A-diffeomorphisms, Inventiones Math., 11 (1970), 99-109. doi: 10.1007/BF01404606.  Google Scholar [12] K. Sigmund, On the distribution of periodic points for $\beta-$shifts, Monatsh. Math, 82 (1976), 247-252. doi: 10.1007/BF01526329.  Google Scholar [13] R. Sturman, and J. Stark, Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity, 13 (2000) 113-143.  Google Scholar [14] Y. Takahashi, Entropy Functional (free energy) for Dynamical Systems and their Random Perturbations, in "Stochastic Analysis North-Holland Math. Library" (Ed. K. It$\hato$), 32 (1982), 437-467.  Google Scholar [15] F. Takens, Heteroclinic attractors: time averages and moduli of topological conjugacy, Bol. Soc. Brasil. Math., 25 (1994), 107-120. doi: 10.1007/BF01232938.  Google Scholar [16] P. Walters, "An Introduction to Ergodic Theory," GTM 79, New York: Springer, 1982.  Google Scholar

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##### References:
 [1] Y. Cao, On growth rates of sub-additive functions for semi-flows: determined and random cases, J. Diff. Eqns, 231 (2006), 1-17. 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), 151-164. 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), 1512-1519. doi: 10.1016/j.automatica.2011.02.034.  Google Scholar [5] X. Dai, Optimal state points of the subadditive ergodic theorem, Nonlinearity, 24 (2011), 1565-1573. 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), 678-689; Translation in Math. Notes, 82 (2007), 608-618.  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, Cambridge: Cambridge University Press, 1995.  Google Scholar [8] J. F. C. Kingman, Subadditive ergodic theory, Ann. Probab., 1 (1973), 883-909.  Google Scholar [9] M. Misiurewicz, Ergodic natural measures, in "Algebraic and Topological Dynamics", volume 385, of Contemp. Math., 1-6 Amer. Math. Soc. Providence R. I. 2005.  Google Scholar [10] S. J. Schreiber, On growth rates of subadditive functions for semi-flows, J. Diff. Eqns, 148 (1998), 334-350. doi: 10.1006/jdeq.1998.3471.  Google Scholar [11] K. Sigmund, Generic properties of invariant measures for axiom A-diffeomorphisms, Inventiones Math., 11 (1970), 99-109. doi: 10.1007/BF01404606.  Google Scholar [12] K. Sigmund, On the distribution of periodic points for $\beta-$shifts, Monatsh. Math, 82 (1976), 247-252. doi: 10.1007/BF01526329.  Google Scholar [13] R. Sturman, and J. Stark, Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity, 13 (2000) 113-143.  Google Scholar [14] Y. Takahashi, Entropy Functional (free energy) for Dynamical Systems and their Random Perturbations, in "Stochastic Analysis North-Holland Math. Library" (Ed. K. It$\hato$), 32 (1982), 437-467.  Google Scholar [15] F. Takens, Heteroclinic attractors: time averages and moduli of topological conjugacy, Bol. Soc. Brasil. Math., 25 (1994), 107-120. doi: 10.1007/BF01232938.  Google Scholar [16] P. Walters, "An Introduction to Ergodic Theory," GTM 79, New York: Springer, 1982.  Google Scholar
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