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, 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

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