• Previous Article
    The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates
  • DCDS Home
  • This Issue
  • Next Article
    On $ n $-tuplewise IP-sensitivity and thick sensitivity
June  2022, 42(6): 2795-2857. doi: 10.3934/dcds.2022001

Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles

School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia

Received  June 2020 Revised  September 2021 Published  June 2022 Early access  January 2022

We develop a random version of the perturbation theory of Gouëzel, Keller, and Liverani, and consequently obtain results on the stability of Oseledets splittings and Lyapunov exponents for operator cocycles. By applying the theory to the Perron-Frobenius operator cocycles associated to random $ \mathcal{C}^k $ expanding maps on $ S^1 $ ($ k \ge 2 $) we provide conditions for the stability of Lyapunov exponents and Oseledets splitting of the cocycle under (ⅰ) uniformly small fiber-wise $ \mathcal{C}^{k-1} $-perturbations to the random dynamics, and (ⅱ) numerical approximation via a Fejér kernel method. A notable addition to our approach is the use of Saks spaces, which allow us to weaken the hypotheses of Gouëzel-Keller-Liverani perturbation theory, provides a unifying framework for key concepts in the so-called 'functional analytic' approach to studying dynamical systems, and has applications to the construction of anisotropic norms adapted to dynamical systems.

Citation: Harry Crimmins. Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2795-2857. doi: 10.3934/dcds.2022001
References:
[1]

A. Alexiewicz, On the two-norm convergence, Studia Math., 14 (1953), 49-56.  doi: 10.4064/sm-14-1-49-56.

[2]

A. Alexiewicz and Z. Semadeni, Linear functionals on two-norm spaces, Studia Math., 17 (1958), 121-140.  doi: 10.4064/sm-17-2-121-140.

[3]

A. Alexiewicz and Z. Semadeni, The two-norm spaces and their conjugate spaces, Studia Math., 18 (1959), 275-293.  doi: 10.4064/sm-18-3-275-293.

[4]

V. Baladi, Correlation spectrum of quenched and annealed equilibrium states for random expanding maps, Comm. Math. Phys., 186 (1997), 671-700.  doi: 10.1007/s002200050124.

[5]

V. Baladi, Positive Transfer Operators and Decay of Correlations, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633.

[6]

V. Baladi, Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, Springer, 2018. doi: 10.1007/978-3-319-77661-3.

[7]

V. BaladiA. Kondah and B. Schmitt, Random correlations for small perturbations of expanding maps, Random Comput. Dynam., 4 (1996), 179-204. 

[8]

M. BlankG. Keller and C. Liverani, Ruelle–Perron–Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973.  doi: 10.1088/0951-7715/15/6/309.

[9]

A. Blumenthal, A volume-based approach to the multiplicative ergodic theorem on Banach spaces, Discrete Contin. Dyn. Syst., 36 (2016), 2377-2403.  doi: 10.3934/dcds.2016.36.2377.

[10]

T. Bogenschütz, Stochastic stability of invariant subspaces, Ergodic Theory Dynam. Systems, 20 (2000), 663-680.  doi: 10.1017/S0143385700000353.

[11]

A. Boyarsky and P. Góra, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension, Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-1-4612-2024-4.

[12] M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, 2002.  doi: 10.1017/CBO9780511755316.
[13]

J. Cooper, Saks Spaces and Applications to Functional Analysis, no. 139 in North-Holland Mathematics Studies, North-Holland, 1987, URL https://books.google.com.au/books?id=JjdPAQAAIAAJ.

[14]

J. Cooper, Saks Spaces and Applications to Functional Analysis, North-Holland Mathematics Studies, Elsevier Science, 2011.

[15]

M. DellnitzG. Froyland and S. Sertl, On the isolated spectrum of the Perron-Frobenius operator, Nonlinearity, 13 (2000), 1171-1188.  doi: 10.1088/0951-7715/13/4/310.

[16]

G. FroylandC. González-Tokman and A. Quas, Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools, J. Comput. Dyn, 1 (2014), 249-278.  doi: 10.3934/jcd.2014.1.249.

[17]

G. FroylandC. González-Tokman and A. Quas, Stability and approximation of random invariant densities for Lasota–Yorke map cocycles, Nonlinearity, 27 (2014), 647-660.  doi: 10.1088/0951-7715/27/4/647.

[18]

G. FroylandS. Lloyd and A. Quas, Coherent structures and isolated spectrum for Perron–Frobenius cocycles, Theory Dynam. Systems, 30 (2010), 729-756.  doi: 10.1017/S0143385709000339.

[19]

G. FroylandS. Lloyd and A. Quas, A semi-invertible Oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst., 33 (2013), 3835-3860.  doi: 10.3934/dcds.2013.33.3835.

[20]

G. FroylandS. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Phys. D, 239 (2010), 1527-1541.  doi: 10.1016/j.physd.2010.03.009.

[21]

S. Galatolo, Statistical properties of dynamics. Introduction to the functional analytic approach, preprint, arXiv: 1510.02615.

[22]

C. González-Tokman, Multiplicative ergodic theorems for transfer operators: Towards the identification and analysis of coherent structures in non-autonomous dynamical systems, Contemp. Math, 709 (2018), 31-52.  doi: 10.1090/conm/709/14290.

[23]

C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem, Ergodic Theory Dynam. Systems, 34 (2014), 1230-1272.  doi: 10.1017/etds.2012.189.

[24]

C. González-Tokman and A. Quas, A concise proof of the multiplicative ergodic theorem on Banach spaces, J. Mod. Dyn., 9 (2015), 237-255.  doi: 10.3934/jmd.2015.9.237.

[25]

C. González-Tokman and A. Quas, Stability and collapse of the Lyapunov spectrum for Perron-Frobenius operator cocycles, J. Eur. Math. Soc, 23 (2021), 3419-3457.  doi: 10.4171/JEMS/1096.

[26]

S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.  doi: 10.1017/S0143385705000374.

[27]

H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634.  doi: 10.2307/2160348.

[28]

T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966.

[29] Y. Katznelson, An Introduction to Harmonic Analysis, 3$^{rd}$ edtion, Cambridge University Press, 2004.  doi: 10.1017/CBO9781139165372.
[30]

G. Keller, Stochastic stability in some chaotic dynamical systems, Monatsh. Math., 94 (1982), 313-333.  doi: 10.1007/BF01667385.

[31]

G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 69 (1985), 461-478.  doi: 10.1007/BF00532744.

[32]

G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 141-152. 

[33]

G. Keller and H. H. Rugh, Eigenfunctions for smooth expanding circle maps, Nonlinearity, 17 (2004), 1723-1730.  doi: 10.1088/0951-7715/17/5/009.

[34]

Z. Lian and K. Lu, Lyapunov Eexponents and Invariant Manifolds for Random Dynamical Systems in A Banach Space, Amer. Math. Soc., (2010).  doi: 10.1090/S0065-9266-10-00574-0.

[35]

C. Liverani, Invariant measures and their properties. A functional analytic point of view, Dynamical Systems. Part Ⅱ, (2003), 185-237. 

[36]

Y. Nakano, Stochastic stability for fiber expanding maps via a perturbative spectral approach, Stoch. Dyn., 16 (2016), 1650011, 15 pp. doi: 10.1142/S0219493716500118.

[37]

M. Novel, p-dimensional cones and applications, preprint, arXiv: 1712.00762.

[38]

V. I. Oseledets, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, rudy Moskov. Mat. Obšč., 19 (1968), 179-210. 

[39]

A. QuasP. Thieullen and M. Zarrabi, Explicit bounds for separation between Oseledets subspaces, Dyn. Syst., 34 (2019), 517-560.  doi: 10.1080/14689367.2019.1571562.

[40]

H. Schaefer, Topological Vector Spaces, Graduate Texts in Mathematics, Vol. 3. Springer-Verlag, New York-Berlin, 1971.

[41]

J. Sedro, Étude de Systèmes Dynamiques Avec Perte de Régularité, PhD thesis, 2018.

[42]

J. SlipantschukO. F. Bandtlow and W. Just, Analytic expanding circle maps with explicit spectra, Nonlinearity, 26 (2013), 3231-3245.  doi: 10.1088/0951-7715/26/12/3231.

[43]

T. Tao, A quick application of the closed graph theorem, https://terrytao.wordpress.com/2016/04/22/a-quick-application-of-the-closed-graph-theorem/#comments, Accessed: 2019-11-21.

[44]

P. Thieullen, Fibrés dynamiques asymptotiquement compacts exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 49-97.  doi: 10.1016/s0294-1449(16)30373-0.

[45]

M. G. Varzaneh and S. Riedel, A dynamical theory for singular stochastic delay differential equations with a Multiplicative Ergodic Theorem on fields of Banach spaces, arXiv preprint, arXiv: 1903.01172.

show all references

References:
[1]

A. Alexiewicz, On the two-norm convergence, Studia Math., 14 (1953), 49-56.  doi: 10.4064/sm-14-1-49-56.

[2]

A. Alexiewicz and Z. Semadeni, Linear functionals on two-norm spaces, Studia Math., 17 (1958), 121-140.  doi: 10.4064/sm-17-2-121-140.

[3]

A. Alexiewicz and Z. Semadeni, The two-norm spaces and their conjugate spaces, Studia Math., 18 (1959), 275-293.  doi: 10.4064/sm-18-3-275-293.

[4]

V. Baladi, Correlation spectrum of quenched and annealed equilibrium states for random expanding maps, Comm. Math. Phys., 186 (1997), 671-700.  doi: 10.1007/s002200050124.

[5]

V. Baladi, Positive Transfer Operators and Decay of Correlations, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633.

[6]

V. Baladi, Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, Springer, 2018. doi: 10.1007/978-3-319-77661-3.

[7]

V. BaladiA. Kondah and B. Schmitt, Random correlations for small perturbations of expanding maps, Random Comput. Dynam., 4 (1996), 179-204. 

[8]

M. BlankG. Keller and C. Liverani, Ruelle–Perron–Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973.  doi: 10.1088/0951-7715/15/6/309.

[9]

A. Blumenthal, A volume-based approach to the multiplicative ergodic theorem on Banach spaces, Discrete Contin. Dyn. Syst., 36 (2016), 2377-2403.  doi: 10.3934/dcds.2016.36.2377.

[10]

T. Bogenschütz, Stochastic stability of invariant subspaces, Ergodic Theory Dynam. Systems, 20 (2000), 663-680.  doi: 10.1017/S0143385700000353.

[11]

A. Boyarsky and P. Góra, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension, Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-1-4612-2024-4.

[12] M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, 2002.  doi: 10.1017/CBO9780511755316.
[13]

J. Cooper, Saks Spaces and Applications to Functional Analysis, no. 139 in North-Holland Mathematics Studies, North-Holland, 1987, URL https://books.google.com.au/books?id=JjdPAQAAIAAJ.

[14]

J. Cooper, Saks Spaces and Applications to Functional Analysis, North-Holland Mathematics Studies, Elsevier Science, 2011.

[15]

M. DellnitzG. Froyland and S. Sertl, On the isolated spectrum of the Perron-Frobenius operator, Nonlinearity, 13 (2000), 1171-1188.  doi: 10.1088/0951-7715/13/4/310.

[16]

G. FroylandC. González-Tokman and A. Quas, Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools, J. Comput. Dyn, 1 (2014), 249-278.  doi: 10.3934/jcd.2014.1.249.

[17]

G. FroylandC. González-Tokman and A. Quas, Stability and approximation of random invariant densities for Lasota–Yorke map cocycles, Nonlinearity, 27 (2014), 647-660.  doi: 10.1088/0951-7715/27/4/647.

[18]

G. FroylandS. Lloyd and A. Quas, Coherent structures and isolated spectrum for Perron–Frobenius cocycles, Theory Dynam. Systems, 30 (2010), 729-756.  doi: 10.1017/S0143385709000339.

[19]

G. FroylandS. Lloyd and A. Quas, A semi-invertible Oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst., 33 (2013), 3835-3860.  doi: 10.3934/dcds.2013.33.3835.

[20]

G. FroylandS. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Phys. D, 239 (2010), 1527-1541.  doi: 10.1016/j.physd.2010.03.009.

[21]

S. Galatolo, Statistical properties of dynamics. Introduction to the functional analytic approach, preprint, arXiv: 1510.02615.

[22]

C. González-Tokman, Multiplicative ergodic theorems for transfer operators: Towards the identification and analysis of coherent structures in non-autonomous dynamical systems, Contemp. Math, 709 (2018), 31-52.  doi: 10.1090/conm/709/14290.

[23]

C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem, Ergodic Theory Dynam. Systems, 34 (2014), 1230-1272.  doi: 10.1017/etds.2012.189.

[24]

C. González-Tokman and A. Quas, A concise proof of the multiplicative ergodic theorem on Banach spaces, J. Mod. Dyn., 9 (2015), 237-255.  doi: 10.3934/jmd.2015.9.237.

[25]

C. González-Tokman and A. Quas, Stability and collapse of the Lyapunov spectrum for Perron-Frobenius operator cocycles, J. Eur. Math. Soc, 23 (2021), 3419-3457.  doi: 10.4171/JEMS/1096.

[26]

S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.  doi: 10.1017/S0143385705000374.

[27]

H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634.  doi: 10.2307/2160348.

[28]

T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966.

[29] Y. Katznelson, An Introduction to Harmonic Analysis, 3$^{rd}$ edtion, Cambridge University Press, 2004.  doi: 10.1017/CBO9781139165372.
[30]

G. Keller, Stochastic stability in some chaotic dynamical systems, Monatsh. Math., 94 (1982), 313-333.  doi: 10.1007/BF01667385.

[31]

G. Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 69 (1985), 461-478.  doi: 10.1007/BF00532744.

[32]

G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 141-152. 

[33]

G. Keller and H. H. Rugh, Eigenfunctions for smooth expanding circle maps, Nonlinearity, 17 (2004), 1723-1730.  doi: 10.1088/0951-7715/17/5/009.

[34]

Z. Lian and K. Lu, Lyapunov Eexponents and Invariant Manifolds for Random Dynamical Systems in A Banach Space, Amer. Math. Soc., (2010).  doi: 10.1090/S0065-9266-10-00574-0.

[35]

C. Liverani, Invariant measures and their properties. A functional analytic point of view, Dynamical Systems. Part Ⅱ, (2003), 185-237. 

[36]

Y. Nakano, Stochastic stability for fiber expanding maps via a perturbative spectral approach, Stoch. Dyn., 16 (2016), 1650011, 15 pp. doi: 10.1142/S0219493716500118.

[37]

M. Novel, p-dimensional cones and applications, preprint, arXiv: 1712.00762.

[38]

V. I. Oseledets, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, rudy Moskov. Mat. Obšč., 19 (1968), 179-210. 

[39]

A. QuasP. Thieullen and M. Zarrabi, Explicit bounds for separation between Oseledets subspaces, Dyn. Syst., 34 (2019), 517-560.  doi: 10.1080/14689367.2019.1571562.

[40]

H. Schaefer, Topological Vector Spaces, Graduate Texts in Mathematics, Vol. 3. Springer-Verlag, New York-Berlin, 1971.

[41]

J. Sedro, Étude de Systèmes Dynamiques Avec Perte de Régularité, PhD thesis, 2018.

[42]

J. SlipantschukO. F. Bandtlow and W. Just, Analytic expanding circle maps with explicit spectra, Nonlinearity, 26 (2013), 3231-3245.  doi: 10.1088/0951-7715/26/12/3231.

[43]

T. Tao, A quick application of the closed graph theorem, https://terrytao.wordpress.com/2016/04/22/a-quick-application-of-the-closed-graph-theorem/#comments, Accessed: 2019-11-21.

[44]

P. Thieullen, Fibrés dynamiques asymptotiquement compacts exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 49-97.  doi: 10.1016/s0294-1449(16)30373-0.

[45]

M. G. Varzaneh and S. Riedel, A dynamical theory for singular stochastic delay differential equations with a Multiplicative Ergodic Theorem on fields of Banach spaces, arXiv preprint, arXiv: 1903.01172.

[1]

Kening Lu, Alexandra Neamţu, Björn Schmalfuss. On the Oseledets-splitting for infinite-dimensional random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1219-1242. doi: 10.3934/dcdsb.2018149

[2]

Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2235-2255. doi: 10.3934/cpaa.2020098

[3]

Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122

[4]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355

[5]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1

[6]

Bixiang Wang. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745

[7]

Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701

[8]

Henri Schurz. Moment attractivity, stability and contractivity exponents of stochastic dynamical systems. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 487-515. doi: 10.3934/dcds.2001.7.487

[9]

Alexandra Rodkina, Henri Schurz, Leonid Shaikhet. Almost sure stability of some stochastic dynamical systems with memory. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 571-593. doi: 10.3934/dcds.2008.21.571

[10]

Yujun Zhu. Preimage entropy for random dynamical systems. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 829-851. doi: 10.3934/dcds.2007.18.829

[11]

Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639

[12]

Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093

[13]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473

[14]

Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727

[15]

Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285

[16]

Weigu Li, Kening Lu. A Siegel theorem for dynamical systems under random perturbations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 635-642. doi: 10.3934/dcdsb.2008.9.635

[17]

Yuri Kifer. Computations in dynamical systems via random perturbations. Discrete and Continuous Dynamical Systems, 1997, 3 (4) : 457-476. doi: 10.3934/dcds.1997.3.457

[18]

Thomas Bogenschütz, Achim Doebler. Large deviations in expanding random dynamical systems. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 805-812. doi: 10.3934/dcds.1999.5.805

[19]

Wenqiang Zhao. Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3395-3438. doi: 10.3934/dcdsb.2018326

[20]

Karl P. Hadeler. Quiescent phases and stability in discrete time dynamical systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 129-152. doi: 10.3934/dcdsb.2015.20.129

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (185)
  • HTML views (129)
  • Cited by (0)

Other articles
by authors

[Back to Top]