An ergodic theory approach to chaos

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[5]	F. Bayart and É. Matheron, Mixing operators and small subsets of the circle,, preprint, (). doi: 10.1515/crelle-2014-0002.
[6]	G. D. Birkhoff, Démonstration d'un théorème élémentaire sur les fonctions entières, C.R. Acad. Sci. Paris, 189 (1929), 473-475.
[7]	C. Blair and L. Rubel, A universal entire function, Amer. Math. Monthly, 90 (1983), 331-332. doi: 10.2307/2975786.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	S. M. Duyos-Ruis, Universal functions of the structure of the space of entire functions, Soviet Math. Dokl., 30 (1984), 713-716.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	E. Hopf, Statistical hydromechanics and functional calculus, J. Rational Mech. Anal., 1 (1952), 87-123.
[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.
[26]	U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics, 6, Walter de Gruyter & Co., Berlin, 1985. doi: 10.1515/9783110844641.
[27]	D. Landers and L. Roggie, An ergodic theorem for Fréchet-valued random variables, Proc. Amer. Math. Soc., 72 (1978), 49-53.
[28]	A. Lasota, Invariant measures and a linear model of turbulence, Rend. Sem. Mat. Univ. Padova, 61 (1979), 40-48.
[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.
[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.
[31]	W. Luh, On universal functions, Colloq. Math. Soc., János Bolyai, 19 (1976), 503-511.
[32]	G. R. MacLane, Sequences of derivatives and normal families, J. Analyse Math., 2 (1952), 72-87. doi: 10.1007/BF02786968.
[33]	H. Méndez-Lango, Is the process of finding $f'$ chaotic?, Rev. Integr. Temas Mat., 22 (2004), 37-41.
[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.
[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.
[36]	M. Pollicott and M. Yuri, Dynamical Systems and Ergodic Theory, London Mathematical Society Student Texts, 40, Cambridge University Press, Cambridge, 1998.
[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.
[38]	S. Rolewicz, On orbits of elements, Studia Math., 32 (1969), 17-22.
[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.
[40]	________, Ergodic properties of hyperbolic systems of partial differential equations, Bull. Polish Acad. Sci. Math., 33 (1985), 595-599.
[41]	________, Ergodic measures on topological spaces, Univ. Iagell. Ac. Math., 26 (1987), 231-237.
[42]	________, An abstract Wiener measure invariant under a partial differential equation, Bull. Polish Acad. Sci. Math., 35 (1987), 289-295.
[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.
[44]	________, Gaussian measure-preserving linear transformations, Univ. Iagell. Ac. Math., 30 (1993), 105-112.
[45]	________, Chaos for some infinite-dimensional dynamical systems, Math. Meth. Appl. Sci., 27 (2004), 723-738. doi: 10.1002/mma.498.
[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.
[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.
[48]	S. Wiggins, Chaotic Transport in Dynamical Systems, Interdisciplinary Applied Mathematics, 2, Springer, New York, 1992. doi: 10.1007/978-1-4757-3896-4.

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.
[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.
[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.
[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.
[5]	F. Bayart and É. Matheron, Mixing operators and small subsets of the circle,, preprint, (). doi: 10.1515/crelle-2014-0002.
[6]	G. D. Birkhoff, Démonstration d'un théorème élémentaire sur les fonctions entières, C.R. Acad. Sci. Paris, 189 (1929), 473-475.
[7]	C. Blair and L. Rubel, A universal entire function, Amer. Math. Monthly, 90 (1983), 331-332. doi: 10.2307/2975786.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	S. M. Duyos-Ruis, Universal functions of the structure of the space of entire functions, Soviet Math. Dokl., 30 (1984), 713-716.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	E. Hopf, Statistical hydromechanics and functional calculus, J. Rational Mech. Anal., 1 (1952), 87-123.
[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.
[26]	U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics, 6, Walter de Gruyter & Co., Berlin, 1985. doi: 10.1515/9783110844641.
[27]	D. Landers and L. Roggie, An ergodic theorem for Fréchet-valued random variables, Proc. Amer. Math. Soc., 72 (1978), 49-53.
[28]	A. Lasota, Invariant measures and a linear model of turbulence, Rend. Sem. Mat. Univ. Padova, 61 (1979), 40-48.
[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.
[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.
[31]	W. Luh, On universal functions, Colloq. Math. Soc., János Bolyai, 19 (1976), 503-511.
[32]	G. R. MacLane, Sequences of derivatives and normal families, J. Analyse Math., 2 (1952), 72-87. doi: 10.1007/BF02786968.
[33]	H. Méndez-Lango, Is the process of finding $f'$ chaotic?, Rev. Integr. Temas Mat., 22 (2004), 37-41.
[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.
[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.
[36]	M. Pollicott and M. Yuri, Dynamical Systems and Ergodic Theory, London Mathematical Society Student Texts, 40, Cambridge University Press, Cambridge, 1998.
[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.
[38]	S. Rolewicz, On orbits of elements, Studia Math., 32 (1969), 17-22.
[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.
[40]	________, Ergodic properties of hyperbolic systems of partial differential equations, Bull. Polish Acad. Sci. Math., 33 (1985), 595-599.
[41]	________, Ergodic measures on topological spaces, Univ. Iagell. Ac. Math., 26 (1987), 231-237.
[42]	________, An abstract Wiener measure invariant under a partial differential equation, Bull. Polish Acad. Sci. Math., 35 (1987), 289-295.
[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.
[44]	________, Gaussian measure-preserving linear transformations, Univ. Iagell. Ac. Math., 30 (1993), 105-112.
[45]	________, Chaos for some infinite-dimensional dynamical systems, Math. Meth. Appl. Sci., 27 (2004), 723-738. doi: 10.1002/mma.498.
[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.
[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.
[48]	S. Wiggins, Chaotic Transport in Dynamical Systems, Interdisciplinary Applied Mathematics, 2, Springer, New York, 1992. doi: 10.1007/978-1-4757-3896-4.

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