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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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 & Continuous Dynamical Systems - A, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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 & Continuous Dynamical Systems - B, 2017, 22 (5) : 1835-1855. doi: 10.3934/dcdsb.2017109

[10]

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 & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020317

[11]

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

[12]

Mohan Mallick, R. Shivaji, Byungjae Son, S. Sundar. Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1295-1304. doi: 10.3934/cpaa.2018062

[13]

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

[14]

Rakesh Nandi, Sujit Kumar Samanta, Chesoong Kim. Analysis of $ D $-$ BMAP/G/1 $ queueing system under $ N $-policy and its cost optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020135

[15]

Juan Dávila, Manuel Del Pino, Catalina Pesce, Juncheng Wei. Blow-up for the 3-dimensional axially symmetric harmonic map flow into $ S^2 $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6913-6943. doi: 10.3934/dcds.2019237

[16]

Joaquim Borges, Cristina Fernández-Córdoba, Roger Ten-Valls. On ${{\mathbb{Z}}}_{p^r}{{\mathbb{Z}}}_{p^s}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 169-179. doi: 10.3934/amc.2018011

[17]

Lingyan Cheng, Ruinan Li, Liming Wu. Exponential convergence in the Wasserstein metric $ W_1 $ for one dimensional diffusions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5131-5148. doi: 10.3934/dcds.2020222

[18]

Lakehal Belarbi. Ricci solitons of the $ \mathbb{H}^{2} \times \mathbb{R} $ Lie group. Electronic Research Archive, 2020, 28 (1) : 157-163. doi: 10.3934/era.2020010

[19]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116

[20]

Wenqiang Zhao. Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2499-2526. doi: 10.3934/dcdsb.2018065

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (54)
  • HTML views (75)
  • Cited by (0)

Other articles
by authors

[Back to Top]