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January  2020, 40(1): 491-508. doi: 10.3934/dcds.2020019

Variational principles of invariance pressures on partitions

Xing-Fu Zhong ,

School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou 510006, China

* Corresponding author: Xing-Fu Zhong

Received  March 2019 Revised  June 2019 Published  October 2019

Fund Project: This work was supported by National Nature Science Funds of China (11771459, 11701584) and project of distinctive innovation of Guangdong University of Foreign Studies(18QN30).

We investigate the relations between Bowen and packing invariance pressures and measure-theoretical lower and upper invariance pressures for invariant partitions of a controlled invariant set respectively. We mainly show that Bowen and packing invariance pressures can be determined via the local lower and upper invariance pressures of probability measures, which are analogues of Billingsley's Theorem for the Hausdorff dimension; and give variational principles between Bowen and packing invariance pressures and measure-theoretical lower and upper invariance pressures under some technical assumptions.

Keywords: Variational principle, measure-theoretic invariance pressure, invariance pressure, Bowen invariance pressure, packing invariance pressure.
Mathematics Subject Classification: Primary: 37A35, 37B40, 37C45; Secondary: 93C55.
Citation: Xing-Fu Zhong. Variational principles of invariance pressures on partitions. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 491-508. doi: 10.3934/dcds.2020019
References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Transactions of the American Mathematical Society, 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.

[2]

F. Colonius, Invariance entropy, quasi-stationary measures and control sets, Discrete & Continuous Dynamical Systems - A, 38 (2018), 2093-2123.  doi: 10.3934/dcds.2018086.

[3]

F. Colonius, Metric invariance entropy and conditionally invariant measures, Ergodic Theory and Dynamical Systems, 38 (2018), 921-939.  doi: 10.1017/etds.2016.72.

[4]

F. Colonius, J. A. N. Cossich and A. J. Santana, Invariance pressure of control sets, SIAM Journal on Control and Optimization, 56 (2018), 4130-4147.  doi: 10.1137/18M1191129.

[5]

F. Colonius and C. Kawan, Invariance entropy for control systems, SIAM Journal on Control and Optimization, 48 (2009), 1701-1721.  doi: 10.1137/080713902.

[6]

F. Colonius, C. Kawan and G. Nair, A note on topological feedback entropy and invariance entropy, Systems & Control Letters, 62 (2013), 377-381.  doi: 10.1016/j.sysconle.2013.01.008.

[7]

F. Colonius, A. J. Santana and J. A. N. Cossich, Bounds for invariance pressure, arXiv: 1904.04768.

[8]

F. Colonius, A. J. Santana and J. A. N. Cossich, Invariance pressure for control systems, Journal of Dynamics and Differential Equations, 31 (2019), 1-23.  doi: 10.1007/s10884-018-9646-2.

[9]

D.-J. Feng and W. Huang, Variational principles for topological entropies of subsets, Journal of Functional Analysis, 263 (2012), 2228-2254.  doi: 10.1016/j.jfa.2012.07.010.

[10]

Y. Huang and X. Zhong, Carathéodory–Pesin structures associated with control systems, Systems & Control Letters, 112 (2018), 36-41.  doi: 10.1016/j.sysconle.2017.12.009.

[11]

C. Kawan, Invariance Entropy for Deterministic Control Systems, An introduction. With a foreword by Fritz Colonius. Lecture Notes in Mathematics, 2089. Springer, Cham, 2013. doi: 10.1007/978-3-319-01288-9.

[12]

A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in lebesgue spaces, Dokl. Akad. Nauk SSSR, 951 (1958), 861-864. 

[13]

J.-H. Ma and Z.-Y. Wen, A Billingsley type theorem for Bowen entropy, Comptes Rendus Mathematique, 346 (2008), 503–507, http://www.sciencedirect.com/science/article/pii/S1631073X08000927. doi: 10.1016/j.crma.2008.03.010.

[14]

G. N. Nair, R. J. Evans, I. M. Y. Mareels and W. Moran, Topological feedback entropy and nonlinear stabilization, IEEE Transactions on Automatic Control, 49 (2004), 1585-1597.  doi: 10.1109/TAC.2004.834105.

[15] Y. B. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, University of Chicago Press, Chicago, 1997.  doi: 10.7208/chicago/9780226662237.001.0001.
[16]

X. Tang, W. C. Cheng and Y. Zhao, Variational principle for topological pressures on subsets, Journal of Mathematical Analysis & Applications, 424 (2015), 1272-1285.  doi: 10.1016/j.jmaa.2014.11.066.

[17]

P. Walters, An Introduction to Ergodic Theory. [Graduate texts in mathematics, Vol. 79], Springer-Verlag, New York, 1982.

[18]

T. Wang, Y. Huang and H. Sun, Measure-theoretic invariance entropy for control systems, SIAM Journal on Control and Optimization, 57 (2019), 310-333.  doi: 10.1137/18M1197862.

[19]

X. Zhong and Y. Huang, Invariance pressure dimensions for control systems, Journal of Dynamics and Differential Equations, (2018), 1–18. doi: 10.1007/s10884-018-9701-z.

show all references

References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Transactions of the American Mathematical Society, 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.

[2]

F. Colonius, Invariance entropy, quasi-stationary measures and control sets, Discrete & Continuous Dynamical Systems - A, 38 (2018), 2093-2123.  doi: 10.3934/dcds.2018086.

[3]

F. Colonius, Metric invariance entropy and conditionally invariant measures, Ergodic Theory and Dynamical Systems, 38 (2018), 921-939.  doi: 10.1017/etds.2016.72.

[4]

F. Colonius, J. A. N. Cossich and A. J. Santana, Invariance pressure of control sets, SIAM Journal on Control and Optimization, 56 (2018), 4130-4147.  doi: 10.1137/18M1191129.

[5]

F. Colonius and C. Kawan, Invariance entropy for control systems, SIAM Journal on Control and Optimization, 48 (2009), 1701-1721.  doi: 10.1137/080713902.

[6]

F. Colonius, C. Kawan and G. Nair, A note on topological feedback entropy and invariance entropy, Systems & Control Letters, 62 (2013), 377-381.  doi: 10.1016/j.sysconle.2013.01.008.

[7]

F. Colonius, A. J. Santana and J. A. N. Cossich, Bounds for invariance pressure, arXiv: 1904.04768.

[8]

F. Colonius, A. J. Santana and J. A. N. Cossich, Invariance pressure for control systems, Journal of Dynamics and Differential Equations, 31 (2019), 1-23.  doi: 10.1007/s10884-018-9646-2.

[9]

D.-J. Feng and W. Huang, Variational principles for topological entropies of subsets, Journal of Functional Analysis, 263 (2012), 2228-2254.  doi: 10.1016/j.jfa.2012.07.010.

[10]

Y. Huang and X. Zhong, Carathéodory–Pesin structures associated with control systems, Systems & Control Letters, 112 (2018), 36-41.  doi: 10.1016/j.sysconle.2017.12.009.

[11]

C. Kawan, Invariance Entropy for Deterministic Control Systems, An introduction. With a foreword by Fritz Colonius. Lecture Notes in Mathematics, 2089. Springer, Cham, 2013. doi: 10.1007/978-3-319-01288-9.

[12]

A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in lebesgue spaces, Dokl. Akad. Nauk SSSR, 951 (1958), 861-864. 

[13]

J.-H. Ma and Z.-Y. Wen, A Billingsley type theorem for Bowen entropy, Comptes Rendus Mathematique, 346 (2008), 503–507, http://www.sciencedirect.com/science/article/pii/S1631073X08000927. doi: 10.1016/j.crma.2008.03.010.

[14]

G. N. Nair, R. J. Evans, I. M. Y. Mareels and W. Moran, Topological feedback entropy and nonlinear stabilization, IEEE Transactions on Automatic Control, 49 (2004), 1585-1597.  doi: 10.1109/TAC.2004.834105.

[15] Y. B. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, University of Chicago Press, Chicago, 1997.  doi: 10.7208/chicago/9780226662237.001.0001.
[16]

X. Tang, W. C. Cheng and Y. Zhao, Variational principle for topological pressures on subsets, Journal of Mathematical Analysis & Applications, 424 (2015), 1272-1285.  doi: 10.1016/j.jmaa.2014.11.066.

[17]

P. Walters, An Introduction to Ergodic Theory. [Graduate texts in mathematics, Vol. 79], Springer-Verlag, New York, 1982.

[18]

T. Wang, Y. Huang and H. Sun, Measure-theoretic invariance entropy for control systems, SIAM Journal on Control and Optimization, 57 (2019), 310-333.  doi: 10.1137/18M1197862.

[19]

X. Zhong and Y. Huang, Invariance pressure dimensions for control systems, Journal of Dynamics and Differential Equations, (2018), 1–18. doi: 10.1007/s10884-018-9701-z.

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