October  2020, 40(10): 5795-5814. doi: 10.3934/dcds.2020246

Entropies of commuting transformations on Hilbert spaces

1. 

School of Mathematics, Northwest University, Xi'an 710127, China

2. 

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

* Corresponding author: Zhiming Li

Received  September 2019 Revised  May 2020 Published  June 2020

Fund Project: The first author is supported by NSFC (No: 11871394) and Natural Science Foundation of Shaanxi Province (2020JC-39), the second author is supported by NSFC (No: 11771118)

By establishing Multiplicative Ergodic Theorem for commutative transformations on a separable infinite dimensional Hilbert space, in this paper, we investigate Pesin's entropy formula and SRB measures of a finitely generated random transformations on such space via its commuting generators. Moreover, as an application, we give a formula of Friedland's entropy for certain $ C^{2} $ $ \mathbb{N}^2 $-actions.

Citation: Zhiming Li, Yujun Zhu. Entropies of commuting transformations on Hilbert spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5795-5814. doi: 10.3934/dcds.2020246
References:
[1]

L. Abromov and V. Rokhlin, The entropy of a skew product of measure-preserving transformations, Amer. Math. Soc. Transl. Ser. 2, 48 (1966), 255-265. 

[2]

J. Bahnmüller and P.-D. Liu, Characterization of measures satisfying the Pesin entropy formula for random dynamical systems, J. Dynam. Differential Equations, 10 (1998), 425-448.  doi: 10.1023/A:1022653229891.

[3]

A. Blumenthal and L.-S. Young, Entropy, volume growth and SRB measures for Banach space mappings, Invent. Math., 207 (2017), 833-893.  doi: 10.1007/s00222-016-0678-0.

[4]

M. Einsiedler and D. Lind, Algebraic $\Bbb Z^d$-actions on entropy rank one, Trans. Amer. Math. Soc., 356 (2004), 1799-1831.  doi: 10.1090/S0002-9947-04-03554-8.

[5]

S. Friedland, Entropy of graphs, semigroups and groups, in Ergodic Theory of Zd Actions (eds. M. Pollicott and K. Schmidt), London Math. Soc. Lecture Note Ser. 228, Cambridge Univ. Press, Cambridge, (1996), 319â€"343. doi: 10.1017/CBO9780511662812.013.

[6]

W. Geller and M. Pollicott, An entropy for $\mathbb{Z} ^2$-actions with finite entropy generators, Fund. Math., 157 (1998), 209-220. 

[7]

H.-Y. Hu, Some ergodic properties of commuting diffeomorphisms, Ergodic Theory Dynam. Systems, 13 (1993), 73-100.  doi: 10.1017/S0143385700007215.

[8]

S. A. Kalikow, $T,{{T}^{-1}}$ Transformation is not loosely Bernoulli, Ann. of Math., 115 (1982), 393-409.  doi: 10.2307/1971397.

[9]

Y. Kifer, Ergodic Theory of Random Transformations, Progress in Probability and Statistics, 10. Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3.

[10]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math., 122 (1985), 509-539.  doi: 10.2307/1971328.

[11]

F. Ledrappier and L.-S. Young, Entropy formula for random transformations, Probab. Theory Related Fields, 80 (1988), 217-240.  doi: 10.1007/BF00356103.

[12]

Z. Li and L. Shu, The metric entropy of random dynamical systems in a Banach space: Ruelle inequality, Ergodic Theory Dynam. Systems, 34 (2014), 594-615.  doi: 10.1017/etds.2012.138.

[13]

Z. Li and L. Shu, The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula, Discrete Contin. Dyn. Syst., 33 (2013), 4123-4155.  doi: 10.3934/dcds.2013.33.4123.

[14]

Z. LianP. Liu and K. Lu, Existence of SRB measures for a class of partially hyperbolic attractors in Banach spaces, Discrete Contin. Dyn. Syst., 37 (2017), 3905-3920.  doi: 10.3934/dcds.2017164.

[15]

Z. LianP. Liu and K. Lu, SRB measures for a class of partially hyperbolic attractors in Hilbert spaces, J. Differential Equations, 261 (2016), 1532-1603.  doi: 10.1016/j.jde.2016.04.006.

[16]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a banach space, Mem. Amer. Math. Soc., 206 (2010), 967. doi: 10.1090/S0065-9266-10-00574-0.

[17]

P.-D. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, Lecture Notes in Mathematics, 1606. Springer-Verlag, Berlin, 1995. doi: 10.1007/BFb0094308.

[18]

P.-D. Liu, Dynamics of random transformations: Smooth ergodic theory, Ergodic Theory Dynam. System, 21 (2001), 1279-1319.  doi: 10.1017/S0143385701001614.

[19]

D. Tang, L. Gu and Z. Li, A remark on stochastic flows in a Hilbert space, J. Dyn. Control Syst., to appear. doi: 10.1007/s10883-020-09481-7.

[20]

P. Thieullen, Entropy and the Hausdorff dimension for infinite-dimensional dynamical systems, J. Dynam. Differential Equations, 4 (1992), 127-159.  doi: 10.1007/BF01048158.

[21]

P. Thieullen, Fibres dynamiques. Entropie et dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 119-146.  doi: 10.1016/S0294-1449(16)30242-6.

[22]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.

[23]

L.-S. Young., Generalizations of SRB measures to nonautonomous, random, and infinite dimensional systems, J. Stat. Phys., 166 (2017), 494-515.  doi: 10.1007/s10955-016-1639-0.

[24]

Y. Zhu, Entropy formula for random $\Bbb{Z}^k$-actions, Trans. Amer. Math. Soc., 369 (2017), 4517-4544.  doi: 10.1090/tran/6798.

[25]

Y. Zhu, A note on two types of Lyapunov exponents and entropies for $\Bbb{Z}^k$-actions, J. Math. Anal. Appl., 461 (2018), 38-50.  doi: 10.1016/j.jmaa.2017.12.067.

show all references

References:
[1]

L. Abromov and V. Rokhlin, The entropy of a skew product of measure-preserving transformations, Amer. Math. Soc. Transl. Ser. 2, 48 (1966), 255-265. 

[2]

J. Bahnmüller and P.-D. Liu, Characterization of measures satisfying the Pesin entropy formula for random dynamical systems, J. Dynam. Differential Equations, 10 (1998), 425-448.  doi: 10.1023/A:1022653229891.

[3]

A. Blumenthal and L.-S. Young, Entropy, volume growth and SRB measures for Banach space mappings, Invent. Math., 207 (2017), 833-893.  doi: 10.1007/s00222-016-0678-0.

[4]

M. Einsiedler and D. Lind, Algebraic $\Bbb Z^d$-actions on entropy rank one, Trans. Amer. Math. Soc., 356 (2004), 1799-1831.  doi: 10.1090/S0002-9947-04-03554-8.

[5]

S. Friedland, Entropy of graphs, semigroups and groups, in Ergodic Theory of Zd Actions (eds. M. Pollicott and K. Schmidt), London Math. Soc. Lecture Note Ser. 228, Cambridge Univ. Press, Cambridge, (1996), 319â€"343. doi: 10.1017/CBO9780511662812.013.

[6]

W. Geller and M. Pollicott, An entropy for $\mathbb{Z} ^2$-actions with finite entropy generators, Fund. Math., 157 (1998), 209-220. 

[7]

H.-Y. Hu, Some ergodic properties of commuting diffeomorphisms, Ergodic Theory Dynam. Systems, 13 (1993), 73-100.  doi: 10.1017/S0143385700007215.

[8]

S. A. Kalikow, $T,{{T}^{-1}}$ Transformation is not loosely Bernoulli, Ann. of Math., 115 (1982), 393-409.  doi: 10.2307/1971397.

[9]

Y. Kifer, Ergodic Theory of Random Transformations, Progress in Probability and Statistics, 10. Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3.

[10]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math., 122 (1985), 509-539.  doi: 10.2307/1971328.

[11]

F. Ledrappier and L.-S. Young, Entropy formula for random transformations, Probab. Theory Related Fields, 80 (1988), 217-240.  doi: 10.1007/BF00356103.

[12]

Z. Li and L. Shu, The metric entropy of random dynamical systems in a Banach space: Ruelle inequality, Ergodic Theory Dynam. Systems, 34 (2014), 594-615.  doi: 10.1017/etds.2012.138.

[13]

Z. Li and L. Shu, The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula, Discrete Contin. Dyn. Syst., 33 (2013), 4123-4155.  doi: 10.3934/dcds.2013.33.4123.

[14]

Z. LianP. Liu and K. Lu, Existence of SRB measures for a class of partially hyperbolic attractors in Banach spaces, Discrete Contin. Dyn. Syst., 37 (2017), 3905-3920.  doi: 10.3934/dcds.2017164.

[15]

Z. LianP. Liu and K. Lu, SRB measures for a class of partially hyperbolic attractors in Hilbert spaces, J. Differential Equations, 261 (2016), 1532-1603.  doi: 10.1016/j.jde.2016.04.006.

[16]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems in a banach space, Mem. Amer. Math. Soc., 206 (2010), 967. doi: 10.1090/S0065-9266-10-00574-0.

[17]

P.-D. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, Lecture Notes in Mathematics, 1606. Springer-Verlag, Berlin, 1995. doi: 10.1007/BFb0094308.

[18]

P.-D. Liu, Dynamics of random transformations: Smooth ergodic theory, Ergodic Theory Dynam. System, 21 (2001), 1279-1319.  doi: 10.1017/S0143385701001614.

[19]

D. Tang, L. Gu and Z. Li, A remark on stochastic flows in a Hilbert space, J. Dyn. Control Syst., to appear. doi: 10.1007/s10883-020-09481-7.

[20]

P. Thieullen, Entropy and the Hausdorff dimension for infinite-dimensional dynamical systems, J. Dynam. Differential Equations, 4 (1992), 127-159.  doi: 10.1007/BF01048158.

[21]

P. Thieullen, Fibres dynamiques. Entropie et dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 119-146.  doi: 10.1016/S0294-1449(16)30242-6.

[22]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.

[23]

L.-S. Young., Generalizations of SRB measures to nonautonomous, random, and infinite dimensional systems, J. Stat. Phys., 166 (2017), 494-515.  doi: 10.1007/s10955-016-1639-0.

[24]

Y. Zhu, Entropy formula for random $\Bbb{Z}^k$-actions, Trans. Amer. Math. Soc., 369 (2017), 4517-4544.  doi: 10.1090/tran/6798.

[25]

Y. Zhu, A note on two types of Lyapunov exponents and entropies for $\Bbb{Z}^k$-actions, J. Math. Anal. Appl., 461 (2018), 38-50.  doi: 10.1016/j.jmaa.2017.12.067.

[1]

Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123

[2]

Jianing Chen, Bixiang Wang. Random attractors of supercritical wave equations driven by infinite-dimensional additive noise on $ \mathbb{R}^n $. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022093

[3]

María Anguiano, Alain Haraux. The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors. Evolution Equations and Control Theory, 2017, 6 (3) : 345-356. doi: 10.3934/eect.2017018

[4]

Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421

[5]

Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from $ {{\mathbb{R}}^{2}} $ to $ {{\mathbb{S}}^{2}} $. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3215-3233. doi: 10.3934/dcds.2019228

[6]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

[7]

Hui Liu, Ling Zhang. Multiplicity of closed Reeb orbits on dynamically convex $ \mathbb{R}P^{2n-1} $ for $ n\geq2 $. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1801-1816. doi: 10.3934/dcds.2021172

[8]

Jorge Garcia Villeda. A computable formula for the class number of the imaginary quadratic field $ \mathbb Q(\sqrt{-p}), \ p = 4n-1 $. Electronic Research Archive, 2021, 29 (6) : 3853-3865. doi: 10.3934/era.2021065

[9]

Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ are inclined. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1131-1143. doi: 10.3934/dcdss.2020067

[10]

Roghayeh Mohammadi Hesari, Mahboubeh Hosseinabadi, Rashid Rezaei, Karim Samei. $\mathbb{F}_{p^{m}}\mathbb{F}_{p^{m}}{[u^2]}$-additive skew cyclic codes of length $2p^s $. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022023

[11]

Wenxian Shen, Shuwen Xue. Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $ \mathbb{R}^{N} $. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2893-2925. doi: 10.3934/dcds.2022003

[12]

Pak Tung Ho. Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry. Communications on Pure and Applied Analysis, 2020, 19 (2) : 715-722. doi: 10.3934/cpaa.2020033

[13]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2635-3652. doi: 10.3934/dcds.2020378

[14]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447

[15]

Genni Fragnelli, Jerome A. Goldstein, Rosa Maria Mininni, Silvia Romanelli. Operators of order 2$ n $ with interior degeneracy. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3417-3426. doi: 10.3934/dcdss.2020128

[16]

Magdalena Foryś-Krawiec, Jiří Kupka, Piotr Oprocha, Xueting Tian. On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1271-1296. doi: 10.3934/dcds.2020317

[17]

Vladimir Chepyzhov, Alexei Ilyin, Sergey Zelik. Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1835-1855. doi: 10.3934/dcdsb.2017109

[18]

Tingting Wu, Jian Gao, Yun Gao, Fang-Wei Fu. $ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 641-657. doi: 10.3934/amc.2018038

[19]

Chunyan Zhao, Chengkui Zhong, Xiangming Zhu. Existence of compact $ \varphi $-attracting sets and estimate of their attractive velocity for infinite-dimensional dynamical systems. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022051

[20]

Valeria Banica, Luis Vega. Singularity formation for the 1-D cubic NLS and the Schrödinger map on $\mathbb S^2$. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1317-1329. doi: 10.3934/cpaa.2018064

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (174)
  • HTML views (77)
  • Cited by (0)

Other articles
by authors

[Back to Top]