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An ergodic theory approach to chaos
1. | Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice |
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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