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Distributionally chaotic families of operators on Fréchet spaces
1. | Dept. Matemàtica Aplicada and IUMPA, Universitat Politècnica de València, València, 46022 |
2. | Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia |
3. | Departamento Matemáticas e I.U.M.A., Universidad de Zaragoza, Zaragoza, Spain |
4. | Instituto Universitario de Matemática Pura y Aplicada, Universitat Politécnica de València, València, Spain |
References:
[1] |
A. A. Albanese, X. Barrachina, E. M. Mangino and A. Peris, Distributional chaos for strongly continuous semigroups of operators, Commun. Pure Appl. Anal., 12 (2013), 2069-2082.
doi: 10.3934/cpaa.2013.12.2069. |
[2] |
A. A. Albanese, J. Bonet and W. J. Ricker, $C_0$-semigroups and mean ergodic operators in a class of Fréchet spaces, J. Math. Anal. Appl., 365 (2010), 142-157.
doi: 10.1016/j.jmaa.2009.10.014. |
[3] |
R. M. Aron, J. B. Seoane-Sepúlveda and A. Weber, Chaos on function spaces, Bull. Austral. Math. Soc., 71 (2005), 411-415.
doi: 10.1017/S0004972700038417. |
[4] |
J. Aroza and A. Peris, Chaotic behaviour of birth-and-death models with proliferation, J. Difference Equ. Appl., 18 (2012), 647-655.
doi: 10.1080/10236198.2011.631535. |
[5] |
J. Banasiak, Birth-and-death type systems with parameter and chaotic dynamics of some linear kinetic models, Z. Anal. Anwendungen, 24 (2005), 675-690.
doi: 10.4171/ZAA/1262. |
[6] |
J. Banasiak and M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for chaos, Discrete Contin. Dyn. Syst., 12 (2005), 959-972.
doi: 10.3934/dcds.2005.12.959. |
[7] |
J. Banasiak and M. Moszyński, Dynamics of birth-and-death processes with proliferation-stability and chaos, Discrete Contin. Dyn. Syst., 29 (2011), 67-79.
doi: 10.3934/dcds.2011.29.67. |
[8] |
X. Barrachina and J. A. Conejero, Devaney chaos and distributional chaos in the solution of certain partial differential equations, Abstr. Appl. Anal., Art. ID 457019, 11. |
[9] |
X. Barrachina, J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Distributional chaos for the forward and backward control traffic model, Linear Algebra Appl., 479 (2015), 202-215.
doi: 10.1016/j.laa.2015.04.010. |
[10] |
X. Barrachina and A. Peris, Distributionally chaotic translation semigroups, J. Difference Equ. Appl., 18 (2012), 751-761.
doi: 10.1080/10236198.2011.625945. |
[11] |
F. Bayart and T. Bermúdez, Semigroups of chaotic operators, Bull. Lond. Math. Soc., 41 (2009), 823-830.
doi: 10.1112/blms/bdp055. |
[12] |
F. Bayart and É. Matheron, Dynamics of Linear Operators, vol. 179 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511581113. |
[13] |
T. Bermúdez, A. Bonilla, F. Martínez-Giménez and A. Peris, Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl., 373 (2011), 83-93. |
[14] |
T. Bermúdez, A. Bonilla and A. Peris, On hypercyclicity and supercyclicity criteria, Bull. Austral. Math. Soc., 70 (2004), 45-54.
doi: 10.1017/S0004972700035802. |
[15] |
T. Bermúdez and V. G. Miller, On operators $T$ such that $f(T)$ is hypercyclic, Extracta Math., 15 (2000), 237-241. |
[16] |
L. Bernal-González and K.-G. Grosse-Erdmann, The hypercyclicity criterion for sequences of operators, Studia Math., 157 (2003), 17-32.
doi: 10.4064/sm157-1-2. |
[17] |
L. Bernal-González, D. Pellegrino and J. B. Seoane-Sepúlveda, Linear subsets of nonlinear sets in topological vector spaces, Bull. Amer. Math. Soc. (N.S.), 51 (2014), 71-130.
doi: 10.1090/S0273-0979-2013-01421-6. |
[18] |
N. C. Bernardes Jr., A. Bonilla, V. Müller and A. Peris, Distributional chaos for linear operators, J. Funct. Anal., 265 (2013), 2143-2163.
doi: 10.1016/j.jfa.2013.06.019. |
[19] |
J. Bès, K. C. Chan and S. M. Seubert, Chaotic unbounded differentiation operators, Integral Equations Operator Theory, 40 (2001), 257-267.
doi: 10.1007/BF01299846. |
[20] |
J. Bès and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal., 167 (1999), 94-112.
doi: 10.1006/jfan.1999.3437. |
[21] |
J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Linear chaos for the quick-thinking-driver model, Semigroup Forum, To appear.
doi: 10.1007/s00233-015-9704-6. |
[22] |
J. A. Conejero and F. Martínez-Giménez, Chaotic differential operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 105 (2011), 423-431.
doi: 10.1007/s13398-011-0026-6. |
[23] |
J. A. Conejero and E. M. Mangino, Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators, Mediterr. J. Math., 7 (2010), 101-109.
doi: 10.1007/s00009-010-0030-7. |
[24] |
J. A. Conejero, V. Müller and A. Peris, Hypercyclic behaviour of operators in a hypercyclic $C_0$-semigroup, J. Funct. Anal., 244 (2007), 342-348.
doi: 10.1016/j.jfa.2006.12.008. |
[25] |
J. A. Conejero and A. Peris, Hypercyclic translation $C_0$-semigroups on complex sectors, Discrete Contin. Dyn. Syst., 25 (2009), 1195-1208.
doi: 10.3934/dcds.2009.25.1195. |
[26] |
R. deLaubenfels, H. Emamirad and K.-G. Grosse-Erdmann, Chaos for semigroups of unbounded operators, Math. Nachr., 261/262 (2003), 47-59.
doi: 10.1002/mana.200310112. |
[27] |
R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, vol. 1570 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1994. |
[28] |
B. Dembart, On the theory of semigroups of operators on locally convex spaces, J. Functional Analysis, 16 (1974), 123-160. |
[29] |
W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems, 17 (1997), 793-819.
doi: 10.1017/S0143385797084976. |
[30] |
J. Duan, X.-C. Fu, P.-D. Liu and A. Manning, A linear chaotic quantum harmonic oscillator, Appl. Math. Lett., 12 (1999), 15-19.
doi: 10.1016/S0893-9659(98)00119-0. |
[31] |
S. El Mourchid, The imaginary point spectrum and hypercyclicity, Semigroup Forum, 73 (2006), 313-316.
doi: 10.1007/s00233-005-0533-x. |
[32] |
H. Emamirad, G. R. Goldstein and J. A. Goldstein, Chaotic solution for the Black-Scholes equation, Proc. Amer. Math. Soc., 140 (2012), 2043-2052. |
[33] |
K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. |
[34] |
L. Frerick, E. Jordá, T. Kalmes and J. Wengenroth, Strongly continuous semigroups on some Fréchet spaces, J. Math. Anal. Appl., 412 (2014), 121-124.
doi: 10.1016/j.jmaa.2013.10.053. |
[35] |
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. |
[36] |
J. A. Goldstein, Some remarks on infinitesimal generators of analytic semigroups, Proc. Amer. Math. Soc., 22 (1969), 91-93. |
[37] |
M. González and F. León-Saavedra, Hypercyclicity for the elements of the commutant of an operator, Integral Equations Operator Theory, 80 (2014), 265-274.
doi: 10.1007/s00020-014-2129-x. |
[38] |
K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos, Universitext, Springer, London, 2011.
doi: 10.1007/978-1-4471-2170-1. |
[39] |
K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 345-381.
doi: 10.1090/S0273-0979-99-00788-0. |
[40] |
G. Herzog, On a universality of the heat equation, Math. Nachr., 188 (1997), 169-171.
doi: 10.1002/mana.19971880110. |
[41] |
L. Ji and A. Weber, Dynamics of the heat semigroup on symmetric spaces, Ergodic Theory Dynam. Systems, 30 (2010), 457-468.
doi: 10.1017/S0143385709000133. |
[42] |
T. Kalmes, Hypercyclic $C_0$-semigroups and evolution families generated by first order differential operators, Proc. Amer. Math. Soc., 137 (2009), 3833-3848.
doi: 10.1090/S0002-9939-09-09955-9. |
[43] |
T. Kōmura, Semigroups of operators in locally convex spaces, Journal of Functional Analysis, 2 (1968), 258-296. |
[44] |
M. Kostić, Generalized Semigroups and Cosine Functions, vol. 23 of Posebna Izdanja [Special Editions], Matematički Institut SANU, Belgrade, 2011. |
[45] |
M. Kostić, Some contributions to the theory of abstract Volterra equations, Int. J. Math. Anal. (Ruse), 5 (2011), 1529-1551. |
[46] |
M. Kostić, Abstract Volterra equations in locally convex spaces, Sci. China Math., 55 (2012), 1797-1825.
doi: 10.1007/s11425-012-4477-9. |
[47] |
M. Kostić, Hypercyclic and chaotic integrated $C$-cosine functions, Filomat, 26 (2012), 1-44.
doi: 10.2298/FIL1201001K. |
[48] |
M. Kostić, Abstract Volterra integro-differential equations: approximation and convergence of resolvent operator families, Numer. Funct. Anal. Optim., 35 (2014), 1579-1606.
doi: 10.1080/01630563.2014.908211. |
[49] |
F. Martínez-Giménez, P. Oprocha and A. Peris, Distributional chaos for backward shifts, J. Math. Anal. Appl., 351 (2009), 607-615.
doi: 10.1016/j.jmaa.2008.10.049. |
[50] |
F. Martínez-Giménez, P. Oprocha and A. Peris, Distributional chaos for operators with full scrambled sets, Math. Z., 274 (2013), 603-612.
doi: 10.1007/s00209-012-1087-8. |
[51] |
Q. Menet, Linear chaos and frequent hypercyclicity, Trans. Amer. Math. Soc., to appear. |
[52] |
G. Metafune, $L_p$-spectrum of Ornstein-Uhlenbeck operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 30 (2001), 97-124. |
[53] |
V. Müller, On the Salas theorem and hypercyclicity of $f(T)$, Integral Equations Operator Theory, 67 (2010), 439-448.
doi: 10.1007/s00020-010-1791-x. |
[54] |
P. Oprocha, A quantum harmonic oscillator and strong chaos, J. Phys. A, 39 (2006), 14559-14565.
doi: 10.1088/0305-4470/39/47/003. |
[55] |
B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754.
doi: 10.2307/2154504. |
[56] |
T. Xiao and J. Liang, Laplace transforms and integrated, regularized semigroups in locally convex spaces, J. Funct. Anal., 148 (1997), 448-479.
doi: 10.1006/jfan.1996.3096. |
[57] |
K. Yosida, Functional Analysis, vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Sixth edition, Springer-Verlag, Berlin-New York, 1980. |
show all references
References:
[1] |
A. A. Albanese, X. Barrachina, E. M. Mangino and A. Peris, Distributional chaos for strongly continuous semigroups of operators, Commun. Pure Appl. Anal., 12 (2013), 2069-2082.
doi: 10.3934/cpaa.2013.12.2069. |
[2] |
A. A. Albanese, J. Bonet and W. J. Ricker, $C_0$-semigroups and mean ergodic operators in a class of Fréchet spaces, J. Math. Anal. Appl., 365 (2010), 142-157.
doi: 10.1016/j.jmaa.2009.10.014. |
[3] |
R. M. Aron, J. B. Seoane-Sepúlveda and A. Weber, Chaos on function spaces, Bull. Austral. Math. Soc., 71 (2005), 411-415.
doi: 10.1017/S0004972700038417. |
[4] |
J. Aroza and A. Peris, Chaotic behaviour of birth-and-death models with proliferation, J. Difference Equ. Appl., 18 (2012), 647-655.
doi: 10.1080/10236198.2011.631535. |
[5] |
J. Banasiak, Birth-and-death type systems with parameter and chaotic dynamics of some linear kinetic models, Z. Anal. Anwendungen, 24 (2005), 675-690.
doi: 10.4171/ZAA/1262. |
[6] |
J. Banasiak and M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for chaos, Discrete Contin. Dyn. Syst., 12 (2005), 959-972.
doi: 10.3934/dcds.2005.12.959. |
[7] |
J. Banasiak and M. Moszyński, Dynamics of birth-and-death processes with proliferation-stability and chaos, Discrete Contin. Dyn. Syst., 29 (2011), 67-79.
doi: 10.3934/dcds.2011.29.67. |
[8] |
X. Barrachina and J. A. Conejero, Devaney chaos and distributional chaos in the solution of certain partial differential equations, Abstr. Appl. Anal., Art. ID 457019, 11. |
[9] |
X. Barrachina, J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Distributional chaos for the forward and backward control traffic model, Linear Algebra Appl., 479 (2015), 202-215.
doi: 10.1016/j.laa.2015.04.010. |
[10] |
X. Barrachina and A. Peris, Distributionally chaotic translation semigroups, J. Difference Equ. Appl., 18 (2012), 751-761.
doi: 10.1080/10236198.2011.625945. |
[11] |
F. Bayart and T. Bermúdez, Semigroups of chaotic operators, Bull. Lond. Math. Soc., 41 (2009), 823-830.
doi: 10.1112/blms/bdp055. |
[12] |
F. Bayart and É. Matheron, Dynamics of Linear Operators, vol. 179 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511581113. |
[13] |
T. Bermúdez, A. Bonilla, F. Martínez-Giménez and A. Peris, Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl., 373 (2011), 83-93. |
[14] |
T. Bermúdez, A. Bonilla and A. Peris, On hypercyclicity and supercyclicity criteria, Bull. Austral. Math. Soc., 70 (2004), 45-54.
doi: 10.1017/S0004972700035802. |
[15] |
T. Bermúdez and V. G. Miller, On operators $T$ such that $f(T)$ is hypercyclic, Extracta Math., 15 (2000), 237-241. |
[16] |
L. Bernal-González and K.-G. Grosse-Erdmann, The hypercyclicity criterion for sequences of operators, Studia Math., 157 (2003), 17-32.
doi: 10.4064/sm157-1-2. |
[17] |
L. Bernal-González, D. Pellegrino and J. B. Seoane-Sepúlveda, Linear subsets of nonlinear sets in topological vector spaces, Bull. Amer. Math. Soc. (N.S.), 51 (2014), 71-130.
doi: 10.1090/S0273-0979-2013-01421-6. |
[18] |
N. C. Bernardes Jr., A. Bonilla, V. Müller and A. Peris, Distributional chaos for linear operators, J. Funct. Anal., 265 (2013), 2143-2163.
doi: 10.1016/j.jfa.2013.06.019. |
[19] |
J. Bès, K. C. Chan and S. M. Seubert, Chaotic unbounded differentiation operators, Integral Equations Operator Theory, 40 (2001), 257-267.
doi: 10.1007/BF01299846. |
[20] |
J. Bès and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal., 167 (1999), 94-112.
doi: 10.1006/jfan.1999.3437. |
[21] |
J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Linear chaos for the quick-thinking-driver model, Semigroup Forum, To appear.
doi: 10.1007/s00233-015-9704-6. |
[22] |
J. A. Conejero and F. Martínez-Giménez, Chaotic differential operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 105 (2011), 423-431.
doi: 10.1007/s13398-011-0026-6. |
[23] |
J. A. Conejero and E. M. Mangino, Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators, Mediterr. J. Math., 7 (2010), 101-109.
doi: 10.1007/s00009-010-0030-7. |
[24] |
J. A. Conejero, V. Müller and A. Peris, Hypercyclic behaviour of operators in a hypercyclic $C_0$-semigroup, J. Funct. Anal., 244 (2007), 342-348.
doi: 10.1016/j.jfa.2006.12.008. |
[25] |
J. A. Conejero and A. Peris, Hypercyclic translation $C_0$-semigroups on complex sectors, Discrete Contin. Dyn. Syst., 25 (2009), 1195-1208.
doi: 10.3934/dcds.2009.25.1195. |
[26] |
R. deLaubenfels, H. Emamirad and K.-G. Grosse-Erdmann, Chaos for semigroups of unbounded operators, Math. Nachr., 261/262 (2003), 47-59.
doi: 10.1002/mana.200310112. |
[27] |
R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, vol. 1570 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1994. |
[28] |
B. Dembart, On the theory of semigroups of operators on locally convex spaces, J. Functional Analysis, 16 (1974), 123-160. |
[29] |
W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems, 17 (1997), 793-819.
doi: 10.1017/S0143385797084976. |
[30] |
J. Duan, X.-C. Fu, P.-D. Liu and A. Manning, A linear chaotic quantum harmonic oscillator, Appl. Math. Lett., 12 (1999), 15-19.
doi: 10.1016/S0893-9659(98)00119-0. |
[31] |
S. El Mourchid, The imaginary point spectrum and hypercyclicity, Semigroup Forum, 73 (2006), 313-316.
doi: 10.1007/s00233-005-0533-x. |
[32] |
H. Emamirad, G. R. Goldstein and J. A. Goldstein, Chaotic solution for the Black-Scholes equation, Proc. Amer. Math. Soc., 140 (2012), 2043-2052. |
[33] |
K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. |
[34] |
L. Frerick, E. Jordá, T. Kalmes and J. Wengenroth, Strongly continuous semigroups on some Fréchet spaces, J. Math. Anal. Appl., 412 (2014), 121-124.
doi: 10.1016/j.jmaa.2013.10.053. |
[35] |
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. |
[36] |
J. A. Goldstein, Some remarks on infinitesimal generators of analytic semigroups, Proc. Amer. Math. Soc., 22 (1969), 91-93. |
[37] |
M. González and F. León-Saavedra, Hypercyclicity for the elements of the commutant of an operator, Integral Equations Operator Theory, 80 (2014), 265-274.
doi: 10.1007/s00020-014-2129-x. |
[38] |
K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos, Universitext, Springer, London, 2011.
doi: 10.1007/978-1-4471-2170-1. |
[39] |
K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 345-381.
doi: 10.1090/S0273-0979-99-00788-0. |
[40] |
G. Herzog, On a universality of the heat equation, Math. Nachr., 188 (1997), 169-171.
doi: 10.1002/mana.19971880110. |
[41] |
L. Ji and A. Weber, Dynamics of the heat semigroup on symmetric spaces, Ergodic Theory Dynam. Systems, 30 (2010), 457-468.
doi: 10.1017/S0143385709000133. |
[42] |
T. Kalmes, Hypercyclic $C_0$-semigroups and evolution families generated by first order differential operators, Proc. Amer. Math. Soc., 137 (2009), 3833-3848.
doi: 10.1090/S0002-9939-09-09955-9. |
[43] |
T. Kōmura, Semigroups of operators in locally convex spaces, Journal of Functional Analysis, 2 (1968), 258-296. |
[44] |
M. Kostić, Generalized Semigroups and Cosine Functions, vol. 23 of Posebna Izdanja [Special Editions], Matematički Institut SANU, Belgrade, 2011. |
[45] |
M. Kostić, Some contributions to the theory of abstract Volterra equations, Int. J. Math. Anal. (Ruse), 5 (2011), 1529-1551. |
[46] |
M. Kostić, Abstract Volterra equations in locally convex spaces, Sci. China Math., 55 (2012), 1797-1825.
doi: 10.1007/s11425-012-4477-9. |
[47] |
M. Kostić, Hypercyclic and chaotic integrated $C$-cosine functions, Filomat, 26 (2012), 1-44.
doi: 10.2298/FIL1201001K. |
[48] |
M. Kostić, Abstract Volterra integro-differential equations: approximation and convergence of resolvent operator families, Numer. Funct. Anal. Optim., 35 (2014), 1579-1606.
doi: 10.1080/01630563.2014.908211. |
[49] |
F. Martínez-Giménez, P. Oprocha and A. Peris, Distributional chaos for backward shifts, J. Math. Anal. Appl., 351 (2009), 607-615.
doi: 10.1016/j.jmaa.2008.10.049. |
[50] |
F. Martínez-Giménez, P. Oprocha and A. Peris, Distributional chaos for operators with full scrambled sets, Math. Z., 274 (2013), 603-612.
doi: 10.1007/s00209-012-1087-8. |
[51] |
Q. Menet, Linear chaos and frequent hypercyclicity, Trans. Amer. Math. Soc., to appear. |
[52] |
G. Metafune, $L_p$-spectrum of Ornstein-Uhlenbeck operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 30 (2001), 97-124. |
[53] |
V. Müller, On the Salas theorem and hypercyclicity of $f(T)$, Integral Equations Operator Theory, 67 (2010), 439-448.
doi: 10.1007/s00020-010-1791-x. |
[54] |
P. Oprocha, A quantum harmonic oscillator and strong chaos, J. Phys. A, 39 (2006), 14559-14565.
doi: 10.1088/0305-4470/39/47/003. |
[55] |
B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754.
doi: 10.2307/2154504. |
[56] |
T. Xiao and J. Liang, Laplace transforms and integrated, regularized semigroups in locally convex spaces, J. Funct. Anal., 148 (1997), 448-479.
doi: 10.1006/jfan.1996.3096. |
[57] |
K. Yosida, Functional Analysis, vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Sixth edition, Springer-Verlag, Berlin-New York, 1980. |
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Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2069-2082. doi: 10.3934/cpaa.2013.12.2069 |
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