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 and 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.

[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.

[3]

E. Catsigeras, Milnor-like attractors, preprint, arXiv:1106.4072v2.

[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.

[5]

X. Dai, Optimal state points of the subadditive ergodic theorem, Nonlinearity, 24 (2011), 1565-1573. doi: 10.1088/0951-7715/24/5/009.

[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.

[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.

[8]

J. F. C. Kingman, Subadditive ergodic theory, Ann. Probab., 1 (1973), 883-909.

[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.

[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.

[11]

K. Sigmund, Generic properties of invariant measures for axiom A-diffeomorphisms, Inventiones Math., 11 (1970), 99-109. doi: 10.1007/BF01404606.

[12]

K. Sigmund, On the distribution of periodic points for $\beta-$shifts, Monatsh. Math, 82 (1976), 247-252. doi: 10.1007/BF01526329.

[13]

R. Sturman, and J. Stark, Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity, 13 (2000) 113-143.

[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.

[15]

F. Takens, Heteroclinic attractors: time averages and moduli of topological conjugacy, Bol. Soc. Brasil. Math., 25 (1994), 107-120. doi: 10.1007/BF01232938.

[16]

P. Walters, "An Introduction to Ergodic Theory," GTM 79, New York: Springer, 1982.

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), 1-17. doi: 10.1016/j.jde.2006.08.016.

[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.

[3]

E. Catsigeras, Milnor-like attractors, preprint, arXiv:1106.4072v2.

[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.

[5]

X. Dai, Optimal state points of the subadditive ergodic theorem, Nonlinearity, 24 (2011), 1565-1573. doi: 10.1088/0951-7715/24/5/009.

[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.

[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.

[8]

J. F. C. Kingman, Subadditive ergodic theory, Ann. Probab., 1 (1973), 883-909.

[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.

[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.

[11]

K. Sigmund, Generic properties of invariant measures for axiom A-diffeomorphisms, Inventiones Math., 11 (1970), 99-109. doi: 10.1007/BF01404606.

[12]

K. Sigmund, On the distribution of periodic points for $\beta-$shifts, Monatsh. Math, 82 (1976), 247-252. doi: 10.1007/BF01526329.

[13]

R. Sturman, and J. Stark, Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity, 13 (2000) 113-143.

[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.

[15]

F. Takens, Heteroclinic attractors: time averages and moduli of topological conjugacy, Bol. Soc. Brasil. Math., 25 (1994), 107-120. doi: 10.1007/BF01232938.

[16]

P. Walters, "An Introduction to Ergodic Theory," GTM 79, New York: Springer, 1982.

[1]

Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial and Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112

[2]

Dominic Veconi. SRB measures of singular hyperbolic attractors. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3415-3430. doi: 10.3934/dcds.2022020

[3]

David Parmenter, Mark Pollicott. Gibbs measures for hyperbolic attractors defined by densities. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3953-3977. doi: 10.3934/dcds.2022038

[4]

Leandro Cioletti, Artur O. Lopes, Manuel Stadlbauer. Ruelle operator for continuous potentials and DLR-Gibbs measures. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4625-4652. doi: 10.3934/dcds.2020195

[5]

Akhtam Dzhalilov, Isabelle Liousse, Dieter Mayer. Singular measures of piecewise smooth circle homeomorphisms with two break points. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 381-403. doi: 10.3934/dcds.2009.24.381

[6]

Youngna Choi. Attractors from one dimensional Lorenz-like maps. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 715-730. doi: 10.3934/dcds.2004.11.715

[7]

Vanderlei Horita, Nivaldo Muniz. Basin problem for Hénon-like attractors in arbitrary dimensions. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 481-504. doi: 10.3934/dcds.2006.15.481

[8]

Norihisa Ikoma. Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 943-966. doi: 10.3934/dcds.2015.35.943

[9]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control and Related Fields, 2021, 11 (3) : 555-578. doi: 10.3934/mcrf.2021012

[10]

Evelyn Herberg, Michael Hinze, Henrik Schumacher. Maximal discrete sparsity in parabolic optimal control with measures. Mathematical Control and Related Fields, 2020, 10 (4) : 735-759. doi: 10.3934/mcrf.2020018

[11]

Rostyslav Kravchenko. The action of finite-state tree automorphisms on Bernoulli measures. Journal of Modern Dynamics, 2010, 4 (3) : 443-451. doi: 10.3934/jmd.2010.4.443

[12]

Peidong Liu, Kening Lu. A note on partially hyperbolic attractors: Entropy conjecture and SRB measures. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 341-352. doi: 10.3934/dcds.2015.35.341

[13]

Zeng Lian, Peidong Liu, Kening Lu. Existence of SRB measures for a class of partially hyperbolic attractors in banach spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3905-3920. doi: 10.3934/dcds.2017164

[14]

Donald Ornstein, Benjamin Weiss. Entropy is the only finitely observable invariant. Journal of Modern Dynamics, 2007, 1 (1) : 93-105. doi: 10.3934/jmd.2007.1.93

[15]

Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (I) --- Basic state solutions. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1091-1116. doi: 10.3934/dcdsb.2004.4.1091

[16]

Xuewei Ju, Desheng Li, Jinqiao Duan. Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1175-1197. doi: 10.3934/dcdsb.2019011

[17]

Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021

[18]

Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control and Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185

[19]

Kazimierz Malanowski, Helmut Maurer. Sensitivity analysis for state constrained optimal control problems. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 241-272. doi: 10.3934/dcds.1998.4.241

[20]

Giulia Cavagnari. Regularity results for a time-optimal control problem in the space of probability measures. Mathematical Control and Related Fields, 2017, 7 (2) : 213-233. doi: 10.3934/mcrf.2017007

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (96)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]