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February  2015, 35(2): 757-770. doi: 10.3934/dcds.2015.35.757

An ergodic theory approach to chaos

Ryszard Rudnicki 1,

1. 

Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice

Received  February 2013 Revised  October 2013 Published  September 2014

This paper is devoted to the ergodic-theoretical approach to chaos, which is based on the existence of invariant mixing measures supported on the whole space. As an example of application of the general theory we prove that there exists an invariant mixing measure with respect to the differentiation operator on the space of entire functions. From this theorem it follows the existence of universal entire functions and other chaotic properties of this transformation.
Keywords: chaos, Invariant measure, differentiation operator., universal functions.
Mathematics Subject Classification: Primary: 37L40; Secondary: 28D10, 30D15, 47A1.
Citation: Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757
References:
[1]

J. Auslander and J. A. Yorke, Interval maps, factors of maps and chaos, Tôhoku Math. J. II. Ser., 32 (1980), 177-188. doi: 10.2748/tmj/1178229634.  Google Scholar

[2]

J. Banasiak and M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for chaos, Discr. Contin. Dyn. Syst., 12 (2005), 959-972. doi: 10.3934/dcds.2005.12.959.  Google Scholar

[3]

J. Bass, Stationary functions and their applications to turbulence, J. Math. Anal. Appl., 47 (1974), 354-399. doi: 10.1016/0022-247X(74)90026-2.  Google Scholar

[4]

F. Bayart and S. Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc., 358 (2006), 5083-5117. doi: 10.1090/S0002-9947-06-04019-0.  Google Scholar

[5]

F. Bayart and É. Matheron, Mixing operators and small subsets of the circle,, preprint, ().  doi: 10.1515/crelle-2014-0002.  Google Scholar

[6]

G. D. Birkhoff, Démonstration d'un théorème élémentaire sur les fonctions entières, C.R. Acad. Sci. Paris, 189 (1929), 473-475. Google Scholar

[7]

C. Blair and L. Rubel, A universal entire function, Amer. Math. Monthly, 90 (1983), 331-332. doi: 10.2307/2975786.  Google Scholar

[8]

O. Blasco, A. Bonilla and K.-G. Grosse-Erdmann, Rate of growth of frequently hypercyclic functions, Proc. Edinb. Math. Soc., 53 (2010), 39-59. doi: 10.1017/S0013091508000564.  Google Scholar

[9]

L. Bernal-González and A. Bonilla, Universality of holomorphic functions bounded on closed sets, J. Math. Anal. Appl., 315 (2006), 302-316. doi: 10.1016/j.jmaa.2005.06.010.  Google Scholar

[10]

N. N. Bogoluboff and N. M. Kriloff, La théorie générale de la measure dans son application à l'étude des systèmes dynamiques de la méchanique non-linéare, Ann. Math., 38 (1937), 65-113. doi: 10.2307/1968511.  Google Scholar

[11]

P. Brunovský and J. Komornik, Ergodicity and exactness of the shift on $C[0,\infty]$ and the semiflow of a first order partial differential equation, J. Math. Anal. Appl., 104 (1984), 235-245. doi: 10.1016/0022-247X(84)90045-3.  Google Scholar

[12]

S. A. Chobanyan, V. I. Tarieladze and N. N. Vakhania, Probability Distributions on Banach Spaces, Kluwer Academic Publ., Dordrecht, 1987. doi: 10.1007/978-94-009-3873-1.  Google Scholar

[13]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Grundlehren der Mathematischen Wissenschaften, 245, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[14]

R. deLaubenfels and H. Emamirad, Chaos for functions of discrete and continuous weighted shift operators, Ergod. Th. Dynam. Sys., 21 (2001), 1411-1427. doi: 10.1017/S0143385701001675.  Google Scholar

[15]

W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynamical Systems, 17 (1997), 793-819. doi: 10.1017/S0143385797084976.  Google Scholar

[16]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edition, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Redwood City, CA, 1989.  Google Scholar

[17]

S. M. Duyos-Ruis, Universal functions of the structure of the space of entire functions, Soviet Math. Dokl., 30 (1984), 713-716. Google Scholar

[18]

S. El Mourchid, G. Metafune, A. Rhandi and J. Voigt, On the chaotic behaviour of size structured cell populations, J. Math. Anal. Appl., 339 (2008), 918-924. doi: 10.1016/j.jmaa.2007.07.034.  Google Scholar

[19]

C. Foiaş, Statistical study of Navier-Stokes equations I, II, Rend. Sem. Mat. Univ. Padova, 48 (1972), 219-348 (1973); ibid. 49 (1973), 9-123.  Google Scholar

[20]

R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc., 100 (1987), 281-288. doi: 10.1090/S0002-9939-1987-0884467-4.  Google Scholar

[21]

G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., 98 (1991), 229-269. doi: 10.1016/0022-1236(91)90078-J.  Google Scholar

[22]

K. G. Grosse-Ederman, On the universal functions of G. R. MacLane, Complex Variables Theory Appl., 15 (1990), 193-196. doi: 10.1080/17476939008814450.  Google Scholar

[23]

H. M. Hilden and L. J. Wallen, Some cyclic and non-cyclic vectors of certain operators, Indiana Univ. Math. J., 23 (1974), 557-565.  Google Scholar

[24]

E. Hopf, Statistical hydromechanics and functional calculus, J. Rational Mech. Anal., 1 (1952), 87-123.  Google Scholar

[25]

K. E. Howard, A size structured model of cell dwarfism, Discr. Contin. Dyn. Syst., 1 (2001), 471-484. doi: 10.3934/dcdsb.2001.1.471.  Google Scholar

[26]

U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics, 6, Walter de Gruyter & Co., Berlin, 1985. doi: 10.1515/9783110844641.  Google Scholar

[27]

D. Landers and L. Roggie, An ergodic theorem for Fréchet-valued random variables, Proc. Amer. Math. Soc., 72 (1978), 49-53.  Google Scholar

[28]

A. Lasota, Invariant measures and a linear model of turbulence, Rend. Sem. Mat. Univ. Padova, 61 (1979), 40-48.  Google Scholar

[29]

A. Lasota and M. C. Mackey, Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, Springer Applied Mathematical Sciences, 97, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[30]

A. Lasota and J. Yorke, On the existence of invariant measures for transformations with strictly turbulent trajectories, Bull. Polish Acad. Sci. Math., 25 (1977), 233-238.  Google Scholar

[31]

W. Luh, On universal functions, Colloq. Math. Soc., János Bolyai, 19 (1976), 503-511.  Google Scholar

[32]

G. R. MacLane, Sequences of derivatives and normal families, J. Analyse Math., 2 (1952), 72-87. doi: 10.1007/BF02786968.  Google Scholar

[33]

H. Méndez-Lango, Is the process of finding $f'$ chaotic?, Rev. Integr. Temas Mat., 22 (2004), 37-41.  Google Scholar

[34]

M. Murillo-Arcila and A. Peris, Strong mixing measures for linear operators and frequent hypercyclicity, J. Math. Anal. Appl., 398 (2013), 462-465. doi: 10.1016/j.jmaa.2012.08.050.  Google Scholar

[35]

J. Myjak and R. Rudnicki, Stability versus chaos for a partial differential equation, Chaos Solitons and Fractals, 14 (2002), 607-612. doi: 10.1016/S0960-0779(01)00190-4.  Google Scholar

[36]

M. Pollicott and M. Yuri, Dynamical Systems and Ergodic Theory, London Mathematical Society Student Texts, 40, Cambridge University Press, Cambridge, 1998.  Google Scholar

[37]

A. Rochlin, Exact endomorphisms of Lebesque spaces, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961) 499-530; Amer. Math. Soc. Transl., 39 (1964), 1-36.  Google Scholar

[38]

S. Rolewicz, On orbits of elements, Studia Math., 32 (1969), 17-22.  Google Scholar

[39]

R. Rudnicki, Invariant measures for the flow of a first-order partial differential equation, Ergodic Theory Dynamical Systems, 8 (1985), 437-443. doi: 10.1017/S0143385700003059.  Google Scholar

[40]

________, Ergodic properties of hyperbolic systems of partial differential equations, Bull. Polish Acad. Sci. Math., 33 (1985), 595-599.  Google Scholar

[41]

________, Ergodic measures on topological spaces, Univ. Iagell. Ac. Math., 26 (1987), 231-237.  Google Scholar

[42]

________, An abstract Wiener measure invariant under a partial differential equation, Bull. Polish Acad. Sci. Math., 35 (1987), 289-295.  Google Scholar

[43]

________, Strong ergodic properties of a first-order partial differential equation, J. Math. Anal. Appl., 133 (1988), 14-26. doi: 10.1016/0022-247X(88)90361-7.  Google Scholar

[44]

________, Gaussian measure-preserving linear transformations, Univ. Iagell. Ac. Math., 30 (1993), 105-112.  Google Scholar

[45]

________, Chaos for some infinite-dimensional dynamical systems, Math. Meth. Appl. Sci., 27 (2004), 723-738. doi: 10.1002/mma.498.  Google Scholar

[46]

________, Chaoticity of the blood cell production system, Chaos: An Interdisciplinary Journal of Nonlinear Science, 19 (2009), 043112, 1-6. doi: 10.1063/1.3258364.  Google Scholar

[47]

________, Chaoticity and invariant measures for a cell population model, J. Math. Anal. Appl., 393 (2012), 151-165. doi: 10.1016/j.jmaa.2012.03.055.  Google Scholar

[48]

S. Wiggins, Chaotic Transport in Dynamical Systems, Interdisciplinary Applied Mathematics, 2, Springer, New York, 1992. doi: 10.1007/978-1-4757-3896-4.  Google Scholar

show all references

References:
[1]

J. Auslander and J. A. Yorke, Interval maps, factors of maps and chaos, Tôhoku Math. J. II. Ser., 32 (1980), 177-188. doi: 10.2748/tmj/1178229634.  Google Scholar

[2]

J. Banasiak and M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for chaos, Discr. Contin. Dyn. Syst., 12 (2005), 959-972. doi: 10.3934/dcds.2005.12.959.  Google Scholar

[3]

J. Bass, Stationary functions and their applications to turbulence, J. Math. Anal. Appl., 47 (1974), 354-399. doi: 10.1016/0022-247X(74)90026-2.  Google Scholar

[4]

F. Bayart and S. Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc., 358 (2006), 5083-5117. doi: 10.1090/S0002-9947-06-04019-0.  Google Scholar

[5]

F. Bayart and É. Matheron, Mixing operators and small subsets of the circle,, preprint, ().  doi: 10.1515/crelle-2014-0002.  Google Scholar

[6]

G. D. Birkhoff, Démonstration d'un théorème élémentaire sur les fonctions entières, C.R. Acad. Sci. Paris, 189 (1929), 473-475. Google Scholar

[7]

C. Blair and L. Rubel, A universal entire function, Amer. Math. Monthly, 90 (1983), 331-332. doi: 10.2307/2975786.  Google Scholar

[8]

O. Blasco, A. Bonilla and K.-G. Grosse-Erdmann, Rate of growth of frequently hypercyclic functions, Proc. Edinb. Math. Soc., 53 (2010), 39-59. doi: 10.1017/S0013091508000564.  Google Scholar

[9]

L. Bernal-González and A. Bonilla, Universality of holomorphic functions bounded on closed sets, J. Math. Anal. Appl., 315 (2006), 302-316. doi: 10.1016/j.jmaa.2005.06.010.  Google Scholar

[10]

N. N. Bogoluboff and N. M. Kriloff, La théorie générale de la measure dans son application à l'étude des systèmes dynamiques de la méchanique non-linéare, Ann. Math., 38 (1937), 65-113. doi: 10.2307/1968511.  Google Scholar

[11]

P. Brunovský and J. Komornik, Ergodicity and exactness of the shift on $C[0,\infty]$ and the semiflow of a first order partial differential equation, J. Math. Anal. Appl., 104 (1984), 235-245. doi: 10.1016/0022-247X(84)90045-3.  Google Scholar

[12]

S. A. Chobanyan, V. I. Tarieladze and N. N. Vakhania, Probability Distributions on Banach Spaces, Kluwer Academic Publ., Dordrecht, 1987. doi: 10.1007/978-94-009-3873-1.  Google Scholar

[13]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Grundlehren der Mathematischen Wissenschaften, 245, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[14]

R. deLaubenfels and H. Emamirad, Chaos for functions of discrete and continuous weighted shift operators, Ergod. Th. Dynam. Sys., 21 (2001), 1411-1427. doi: 10.1017/S0143385701001675.  Google Scholar

[15]

W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynamical Systems, 17 (1997), 793-819. doi: 10.1017/S0143385797084976.  Google Scholar

[16]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edition, Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Redwood City, CA, 1989.  Google Scholar

[17]

S. M. Duyos-Ruis, Universal functions of the structure of the space of entire functions, Soviet Math. Dokl., 30 (1984), 713-716. Google Scholar

[18]

S. El Mourchid, G. Metafune, A. Rhandi and J. Voigt, On the chaotic behaviour of size structured cell populations, J. Math. Anal. Appl., 339 (2008), 918-924. doi: 10.1016/j.jmaa.2007.07.034.  Google Scholar

[19]

C. Foiaş, Statistical study of Navier-Stokes equations I, II, Rend. Sem. Mat. Univ. Padova, 48 (1972), 219-348 (1973); ibid. 49 (1973), 9-123.  Google Scholar

[20]

R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc., 100 (1987), 281-288. doi: 10.1090/S0002-9939-1987-0884467-4.  Google Scholar

[21]

G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., 98 (1991), 229-269. doi: 10.1016/0022-1236(91)90078-J.  Google Scholar

[22]

K. G. Grosse-Ederman, On the universal functions of G. R. MacLane, Complex Variables Theory Appl., 15 (1990), 193-196. doi: 10.1080/17476939008814450.  Google Scholar

[23]

H. M. Hilden and L. J. Wallen, Some cyclic and non-cyclic vectors of certain operators, Indiana Univ. Math. J., 23 (1974), 557-565.  Google Scholar

[24]

E. Hopf, Statistical hydromechanics and functional calculus, J. Rational Mech. Anal., 1 (1952), 87-123.  Google Scholar

[25]

K. E. Howard, A size structured model of cell dwarfism, Discr. Contin. Dyn. Syst., 1 (2001), 471-484. doi: 10.3934/dcdsb.2001.1.471.  Google Scholar

[26]

U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics, 6, Walter de Gruyter & Co., Berlin, 1985. doi: 10.1515/9783110844641.  Google Scholar

[27]

D. Landers and L. Roggie, An ergodic theorem for Fréchet-valued random variables, Proc. Amer. Math. Soc., 72 (1978), 49-53.  Google Scholar

[28]

A. Lasota, Invariant measures and a linear model of turbulence, Rend. Sem. Mat. Univ. Padova, 61 (1979), 40-48.  Google Scholar

[29]

A. Lasota and M. C. Mackey, Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, Springer Applied Mathematical Sciences, 97, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[30]

A. Lasota and J. Yorke, On the existence of invariant measures for transformations with strictly turbulent trajectories, Bull. Polish Acad. Sci. Math., 25 (1977), 233-238.  Google Scholar

[31]

W. Luh, On universal functions, Colloq. Math. Soc., János Bolyai, 19 (1976), 503-511.  Google Scholar

[32]

G. R. MacLane, Sequences of derivatives and normal families, J. Analyse Math., 2 (1952), 72-87. doi: 10.1007/BF02786968.  Google Scholar

[33]

H. Méndez-Lango, Is the process of finding $f'$ chaotic?, Rev. Integr. Temas Mat., 22 (2004), 37-41.  Google Scholar

[34]

M. Murillo-Arcila and A. Peris, Strong mixing measures for linear operators and frequent hypercyclicity, J. Math. Anal. Appl., 398 (2013), 462-465. doi: 10.1016/j.jmaa.2012.08.050.  Google Scholar

[35]

J. Myjak and R. Rudnicki, Stability versus chaos for a partial differential equation, Chaos Solitons and Fractals, 14 (2002), 607-612. doi: 10.1016/S0960-0779(01)00190-4.  Google Scholar

[36]

M. Pollicott and M. Yuri, Dynamical Systems and Ergodic Theory, London Mathematical Society Student Texts, 40, Cambridge University Press, Cambridge, 1998.  Google Scholar

[37]

A. Rochlin, Exact endomorphisms of Lebesque spaces, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961) 499-530; Amer. Math. Soc. Transl., 39 (1964), 1-36.  Google Scholar

[38]

S. Rolewicz, On orbits of elements, Studia Math., 32 (1969), 17-22.  Google Scholar

[39]

R. Rudnicki, Invariant measures for the flow of a first-order partial differential equation, Ergodic Theory Dynamical Systems, 8 (1985), 437-443. doi: 10.1017/S0143385700003059.  Google Scholar

[40]

________, Ergodic properties of hyperbolic systems of partial differential equations, Bull. Polish Acad. Sci. Math., 33 (1985), 595-599.  Google Scholar

[41]

________, Ergodic measures on topological spaces, Univ. Iagell. Ac. Math., 26 (1987), 231-237.  Google Scholar

[42]

________, An abstract Wiener measure invariant under a partial differential equation, Bull. Polish Acad. Sci. Math., 35 (1987), 289-295.  Google Scholar

[43]

________, Strong ergodic properties of a first-order partial differential equation, J. Math. Anal. Appl., 133 (1988), 14-26. doi: 10.1016/0022-247X(88)90361-7.  Google Scholar

[44]

________, Gaussian measure-preserving linear transformations, Univ. Iagell. Ac. Math., 30 (1993), 105-112.  Google Scholar

[45]

________, Chaos for some infinite-dimensional dynamical systems, Math. Meth. Appl. Sci., 27 (2004), 723-738. doi: 10.1002/mma.498.  Google Scholar

[46]

________, Chaoticity of the blood cell production system, Chaos: An Interdisciplinary Journal of Nonlinear Science, 19 (2009), 043112, 1-6. doi: 10.1063/1.3258364.  Google Scholar

[47]

________, Chaoticity and invariant measures for a cell population model, J. Math. Anal. Appl., 393 (2012), 151-165. doi: 10.1016/j.jmaa.2012.03.055.  Google Scholar

[48]

S. Wiggins, Chaotic Transport in Dynamical Systems, Interdisciplinary Applied Mathematics, 2, Springer, New York, 1992. doi: 10.1007/978-1-4757-3896-4.  Google Scholar

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