September  2016, 15(5): 1915-1939. doi: 10.3934/cpaa.2016022

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

Received  June 2015 Revised  May 2016 Published  July 2016

The existence of distributional chaos and distributional irregular vectors has been recently considered in the study of linear dynamics of operators and $C_0$-semigroups. In this paper we extend some previous results on both notions to sequences of operators, $C_0$-semigroups, $C$-regularized semigroups, and $\alpha$-times integrated semigroups on Fréchet spaces. We also add a study of rescaled distributionally chaotic $C_0$-semigroups. Some examples are provided to illustrate all these results.
Citation: J. Alberto Conejero, Marko Kostić, Pedro J. Miana, Marina Murillo-Arcila. Distributionally chaotic families of operators on Fréchet spaces. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1915-1939. doi: 10.3934/cpaa.2016022
References:
[1]

A. A. Albanese, X. Barrachina, E. M. Mangino and A. Peris, Distributional chaos for strongly continuous semigroups of operators,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 2069.  doi: 10.3934/cpaa.2013.12.2069.  Google Scholar

[2]

A. A. Albanese, J. Bonet and W. J. Ricker, $C_0$-semigroups and mean ergodic operators in a class of Fréchet spaces,, \emph{J. Math. Anal. Appl.}, 365 (2010), 142.  doi: 10.1016/j.jmaa.2009.10.014.  Google Scholar

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[6]

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

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J. Banasiak and M. Moszyński, Dynamics of birth-and-death processes with proliferation-stability and chaos,, \emph{Discrete Contin. Dyn. Syst.}, 29 (2011), 67.  doi: 10.3934/dcds.2011.29.67.  Google Scholar

[8]

X. Barrachina and J. A. Conejero, Devaney chaos and distributional chaos in the solution of certain partial differential equations,, \emph{Abstr. Appl. Anal.}, (4570).   Google Scholar

[9]

X. Barrachina, J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Distributional chaos for the forward and backward control traffic model,, \emph{Linear Algebra Appl.}, 479 (2015), 202.  doi: 10.1016/j.laa.2015.04.010.  Google Scholar

[10]

X. Barrachina and A. Peris, Distributionally chaotic translation semigroups,, \emph{J. Difference Equ. Appl.}, 18 (2012), 751.  doi: 10.1080/10236198.2011.625945.  Google Scholar

[11]

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[14]

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[20]

J. Bès and A. Peris, Hereditarily hypercyclic operators,, \emph{J. Funct. Anal.}, 167 (1999), 94.  doi: 10.1006/jfan.1999.3437.  Google Scholar

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J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Linear chaos for the quick-thinking-driver model,, \emph{Semigroup Forum}, ().  doi: 10.1007/s00233-015-9704-6.  Google Scholar

[22]

J. A. Conejero and F. Martínez-Giménez, Chaotic differential operators,, \emph{Rev. R. Acad. Cienc. Exactas F\'\i s. Nat. Ser. A Math. RACSAM}, 105 (2011), 423.  doi: 10.1007/s13398-011-0026-6.  Google Scholar

[23]

J. A. Conejero and E. M. Mangino, Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators,, \emph{Mediterr. J. Math.}, 7 (2010), 101.  doi: 10.1007/s00009-010-0030-7.  Google Scholar

[24]

J. A. Conejero, V. Müller and A. Peris, Hypercyclic behaviour of operators in a hypercyclic $C_0$-semigroup,, \emph{J. Funct. Anal.}, 244 (2007), 342.  doi: 10.1016/j.jfa.2006.12.008.  Google Scholar

[25]

J. A. Conejero and A. Peris, Hypercyclic translation $C_0$-semigroups on complex sectors,, \emph{Discrete Contin. Dyn. Syst.}, 25 (2009), 1195.  doi: 10.3934/dcds.2009.25.1195.  Google Scholar

[26]

R. deLaubenfels, H. Emamirad and K.-G. Grosse-Erdmann, Chaos for semigroups of unbounded operators,, \emph{Math. Nachr.}, 261/262 (2003), 47.  doi: 10.1002/mana.200310112.  Google Scholar

[27]

R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, vol. 1570 of Lecture Notes in Mathematics,, Springer-Verlag, (1994).   Google Scholar

[28]

B. Dembart, On the theory of semigroups of operators on locally convex spaces,, \emph{J. Functional Analysis}, 16 (1974), 123.   Google Scholar

[29]

W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators,, \emph{Ergodic Theory Dynam. Systems}, 17 (1997), 793.  doi: 10.1017/S0143385797084976.  Google Scholar

[30]

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[31]

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[32]

H. Emamirad, G. R. Goldstein and J. A. Goldstein, Chaotic solution for the Black-Scholes equation,, \emph{Proc. Amer. Math. Soc.}, 140 (2012), 2043.   Google Scholar

[33]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics,, Springer-Verlag, (2000).   Google Scholar

[34]

L. Frerick, E. Jordá, T. Kalmes and J. Wengenroth, Strongly continuous semigroups on some Fréchet spaces,, \emph{J. Math. Anal. Appl.}, 412 (2014), 121.  doi: 10.1016/j.jmaa.2013.10.053.  Google Scholar

[35]

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

[36]

J. A. Goldstein, Some remarks on infinitesimal generators of analytic semigroups,, \emph{Proc. Amer. Math. Soc.}, 22 (1969), 91.   Google Scholar

[37]

M. González and F. León-Saavedra, Hypercyclicity for the elements of the commutant of an operator,, \emph{Integral Equations Operator Theory}, 80 (2014), 265.  doi: 10.1007/s00020-014-2129-x.  Google Scholar

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K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos,, Universitext, (2011).  doi: 10.1007/978-1-4471-2170-1.  Google Scholar

[39]

K.-G. Grosse-Erdmann, Universal families and hypercyclic operators,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 36 (1999), 345.  doi: 10.1090/S0273-0979-99-00788-0.  Google Scholar

[40]

G. Herzog, On a universality of the heat equation,, \emph{Math. Nachr.}, 188 (1997), 169.  doi: 10.1002/mana.19971880110.  Google Scholar

[41]

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[42]

T. Kalmes, Hypercyclic $C_0$-semigroups and evolution families generated by first order differential operators,, \emph{Proc. Amer. Math. Soc.}, 137 (2009), 3833.  doi: 10.1090/S0002-9939-09-09955-9.  Google Scholar

[43]

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[44]

M. Kostić, Generalized Semigroups and Cosine Functions, vol. 23 of Posebna Izdanja [Special Editions],, Matemati\v cki Institut SANU, (2011).   Google Scholar

[45]

M. Kostić, Some contributions to the theory of abstract Volterra equations,, \emph{Int. J. Math. Anal. (Ruse)}, 5 (2011), 1529.   Google Scholar

[46]

M. Kostić, Abstract Volterra equations in locally convex spaces,, \emph{Sci. China Math.}, 55 (2012), 1797.  doi: 10.1007/s11425-012-4477-9.  Google Scholar

[47]

M. Kostić, Hypercyclic and chaotic integrated $C$-cosine functions,, \emph{Filomat}, 26 (2012), 1.  doi: 10.2298/FIL1201001K.  Google Scholar

[48]

M. Kostić, Abstract Volterra integro-differential equations: approximation and convergence of resolvent operator families,, \emph{Numer. Funct. Anal. Optim.}, 35 (2014), 1579.  doi: 10.1080/01630563.2014.908211.  Google Scholar

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F. Martínez-Giménez, P. Oprocha and A. Peris, Distributional chaos for backward shifts,, \emph{J. Math. Anal. Appl.}, 351 (2009), 607.  doi: 10.1016/j.jmaa.2008.10.049.  Google Scholar

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F. Martínez-Giménez, P. Oprocha and A. Peris, Distributional chaos for operators with full scrambled sets,, \emph{Math. Z.}, 274 (2013), 603.  doi: 10.1007/s00209-012-1087-8.  Google Scholar

[51]

Q. Menet, Linear chaos and frequent hypercyclicity,, \emph{Trans. Amer. Math. Soc.}, ().   Google Scholar

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G. Metafune, $L_p$-spectrum of Ornstein-Uhlenbeck operators,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)}, 30 (2001), 97.   Google Scholar

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V. Müller, On the Salas theorem and hypercyclicity of $f(T)$,, \emph{Integral Equations Operator Theory}, 67 (2010), 439.  doi: 10.1007/s00020-010-1791-x.  Google Scholar

[54]

P. Oprocha, A quantum harmonic oscillator and strong chaos,, \emph{J. Phys. A}, 39 (2006), 14559.  doi: 10.1088/0305-4470/39/47/003.  Google Scholar

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B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval,, \emph{Trans. Amer. Math. Soc.}, 344 (1994), 737.  doi: 10.2307/2154504.  Google Scholar

[56]

T. Xiao and J. Liang, Laplace transforms and integrated, regularized semigroups in locally convex spaces,, \emph{J. Funct. Anal.}, 148 (1997), 448.  doi: 10.1006/jfan.1996.3096.  Google Scholar

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show all references

References:
[1]

A. A. Albanese, X. Barrachina, E. M. Mangino and A. Peris, Distributional chaos for strongly continuous semigroups of operators,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 2069.  doi: 10.3934/cpaa.2013.12.2069.  Google Scholar

[2]

A. A. Albanese, J. Bonet and W. J. Ricker, $C_0$-semigroups and mean ergodic operators in a class of Fréchet spaces,, \emph{J. Math. Anal. Appl.}, 365 (2010), 142.  doi: 10.1016/j.jmaa.2009.10.014.  Google Scholar

[3]

R. M. Aron, J. B. Seoane-Sepúlveda and A. Weber, Chaos on function spaces,, \emph{Bull. Austral. Math. Soc.}, 71 (2005), 411.  doi: 10.1017/S0004972700038417.  Google Scholar

[4]

J. Aroza and A. Peris, Chaotic behaviour of birth-and-death models with proliferation,, \emph{J. Difference Equ. Appl.}, 18 (2012), 647.  doi: 10.1080/10236198.2011.631535.  Google Scholar

[5]

J. Banasiak, Birth-and-death type systems with parameter and chaotic dynamics of some linear kinetic models,, \emph{Z. Anal. Anwendungen}, 24 (2005), 675.  doi: 10.4171/ZAA/1262.  Google Scholar

[6]

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

[7]

J. Banasiak and M. Moszyński, Dynamics of birth-and-death processes with proliferation-stability and chaos,, \emph{Discrete Contin. Dyn. Syst.}, 29 (2011), 67.  doi: 10.3934/dcds.2011.29.67.  Google Scholar

[8]

X. Barrachina and J. A. Conejero, Devaney chaos and distributional chaos in the solution of certain partial differential equations,, \emph{Abstr. Appl. Anal.}, (4570).   Google Scholar

[9]

X. Barrachina, J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Distributional chaos for the forward and backward control traffic model,, \emph{Linear Algebra Appl.}, 479 (2015), 202.  doi: 10.1016/j.laa.2015.04.010.  Google Scholar

[10]

X. Barrachina and A. Peris, Distributionally chaotic translation semigroups,, \emph{J. Difference Equ. Appl.}, 18 (2012), 751.  doi: 10.1080/10236198.2011.625945.  Google Scholar

[11]

F. Bayart and T. Bermúdez, Semigroups of chaotic operators,, \emph{Bull. Lond. Math. Soc.}, 41 (2009), 823.  doi: 10.1112/blms/bdp055.  Google Scholar

[12]

F. Bayart and É. Matheron, Dynamics of Linear Operators, vol. 179 of Cambridge Tracts in Mathematics,, Cambridge University Press, (2009).  doi: 10.1017/CBO9780511581113.  Google Scholar

[13]

T. Bermúdez, A. Bonilla, F. Martínez-Giménez and A. Peris, Li-Yorke and distributionally chaotic operators,, \emph{J. Math. Anal. Appl.}, 373 (2011), 83.   Google Scholar

[14]

T. Bermúdez, A. Bonilla and A. Peris, On hypercyclicity and supercyclicity criteria,, \emph{Bull. Austral. Math. Soc.}, 70 (2004), 45.  doi: 10.1017/S0004972700035802.  Google Scholar

[15]

T. Bermúdez and V. G. Miller, On operators $T$ such that $f(T)$ is hypercyclic,, \emph{Extracta Math.}, 15 (2000), 237.   Google Scholar

[16]

L. Bernal-González and K.-G. Grosse-Erdmann, The hypercyclicity criterion for sequences of operators,, \emph{Studia Math.}, 157 (2003), 17.  doi: 10.4064/sm157-1-2.  Google Scholar

[17]

L. Bernal-González, D. Pellegrino and J. B. Seoane-Sepúlveda, Linear subsets of nonlinear sets in topological vector spaces,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 51 (2014), 71.  doi: 10.1090/S0273-0979-2013-01421-6.  Google Scholar

[18]

N. C. Bernardes Jr., A. Bonilla, V. Müller and A. Peris, Distributional chaos for linear operators,, \emph{J. Funct. Anal.}, 265 (2013), 2143.  doi: 10.1016/j.jfa.2013.06.019.  Google Scholar

[19]

J. Bès, K. C. Chan and S. M. Seubert, Chaotic unbounded differentiation operators,, \emph{Integral Equations Operator Theory}, 40 (2001), 257.  doi: 10.1007/BF01299846.  Google Scholar

[20]

J. Bès and A. Peris, Hereditarily hypercyclic operators,, \emph{J. Funct. Anal.}, 167 (1999), 94.  doi: 10.1006/jfan.1999.3437.  Google Scholar

[21]

J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Linear chaos for the quick-thinking-driver model,, \emph{Semigroup Forum}, ().  doi: 10.1007/s00233-015-9704-6.  Google Scholar

[22]

J. A. Conejero and F. Martínez-Giménez, Chaotic differential operators,, \emph{Rev. R. Acad. Cienc. Exactas F\'\i s. Nat. Ser. A Math. RACSAM}, 105 (2011), 423.  doi: 10.1007/s13398-011-0026-6.  Google Scholar

[23]

J. A. Conejero and E. M. Mangino, Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators,, \emph{Mediterr. J. Math.}, 7 (2010), 101.  doi: 10.1007/s00009-010-0030-7.  Google Scholar

[24]

J. A. Conejero, V. Müller and A. Peris, Hypercyclic behaviour of operators in a hypercyclic $C_0$-semigroup,, \emph{J. Funct. Anal.}, 244 (2007), 342.  doi: 10.1016/j.jfa.2006.12.008.  Google Scholar

[25]

J. A. Conejero and A. Peris, Hypercyclic translation $C_0$-semigroups on complex sectors,, \emph{Discrete Contin. Dyn. Syst.}, 25 (2009), 1195.  doi: 10.3934/dcds.2009.25.1195.  Google Scholar

[26]

R. deLaubenfels, H. Emamirad and K.-G. Grosse-Erdmann, Chaos for semigroups of unbounded operators,, \emph{Math. Nachr.}, 261/262 (2003), 47.  doi: 10.1002/mana.200310112.  Google Scholar

[27]

R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, vol. 1570 of Lecture Notes in Mathematics,, Springer-Verlag, (1994).   Google Scholar

[28]

B. Dembart, On the theory of semigroups of operators on locally convex spaces,, \emph{J. Functional Analysis}, 16 (1974), 123.   Google Scholar

[29]

W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators,, \emph{Ergodic Theory Dynam. Systems}, 17 (1997), 793.  doi: 10.1017/S0143385797084976.  Google Scholar

[30]

J. Duan, X.-C. Fu, P.-D. Liu and A. Manning, A linear chaotic quantum harmonic oscillator,, \emph{Appl. Math. Lett.}, 12 (1999), 15.  doi: 10.1016/S0893-9659(98)00119-0.  Google Scholar

[31]

S. El Mourchid, The imaginary point spectrum and hypercyclicity,, \emph{Semigroup Forum}, 73 (2006), 313.  doi: 10.1007/s00233-005-0533-x.  Google Scholar

[32]

H. Emamirad, G. R. Goldstein and J. A. Goldstein, Chaotic solution for the Black-Scholes equation,, \emph{Proc. Amer. Math. Soc.}, 140 (2012), 2043.   Google Scholar

[33]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics,, Springer-Verlag, (2000).   Google Scholar

[34]

L. Frerick, E. Jordá, T. Kalmes and J. Wengenroth, Strongly continuous semigroups on some Fréchet spaces,, \emph{J. Math. Anal. Appl.}, 412 (2014), 121.  doi: 10.1016/j.jmaa.2013.10.053.  Google Scholar

[35]

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

[36]

J. A. Goldstein, Some remarks on infinitesimal generators of analytic semigroups,, \emph{Proc. Amer. Math. Soc.}, 22 (1969), 91.   Google Scholar

[37]

M. González and F. León-Saavedra, Hypercyclicity for the elements of the commutant of an operator,, \emph{Integral Equations Operator Theory}, 80 (2014), 265.  doi: 10.1007/s00020-014-2129-x.  Google Scholar

[38]

K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos,, Universitext, (2011).  doi: 10.1007/978-1-4471-2170-1.  Google Scholar

[39]

K.-G. Grosse-Erdmann, Universal families and hypercyclic operators,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 36 (1999), 345.  doi: 10.1090/S0273-0979-99-00788-0.  Google Scholar

[40]

G. Herzog, On a universality of the heat equation,, \emph{Math. Nachr.}, 188 (1997), 169.  doi: 10.1002/mana.19971880110.  Google Scholar

[41]

L. Ji and A. Weber, Dynamics of the heat semigroup on symmetric spaces,, \emph{Ergodic Theory Dynam. Systems}, 30 (2010), 457.  doi: 10.1017/S0143385709000133.  Google Scholar

[42]

T. Kalmes, Hypercyclic $C_0$-semigroups and evolution families generated by first order differential operators,, \emph{Proc. Amer. Math. Soc.}, 137 (2009), 3833.  doi: 10.1090/S0002-9939-09-09955-9.  Google Scholar

[43]

T. Kōmura, Semigroups of operators in locally convex spaces,, \emph{Journal of Functional Analysis}, 2 (1968), 258.   Google Scholar

[44]

M. Kostić, Generalized Semigroups and Cosine Functions, vol. 23 of Posebna Izdanja [Special Editions],, Matemati\v cki Institut SANU, (2011).   Google Scholar

[45]

M. Kostić, Some contributions to the theory of abstract Volterra equations,, \emph{Int. J. Math. Anal. (Ruse)}, 5 (2011), 1529.   Google Scholar

[46]

M. Kostić, Abstract Volterra equations in locally convex spaces,, \emph{Sci. China Math.}, 55 (2012), 1797.  doi: 10.1007/s11425-012-4477-9.  Google Scholar

[47]

M. Kostić, Hypercyclic and chaotic integrated $C$-cosine functions,, \emph{Filomat}, 26 (2012), 1.  doi: 10.2298/FIL1201001K.  Google Scholar

[48]

M. Kostić, Abstract Volterra integro-differential equations: approximation and convergence of resolvent operator families,, \emph{Numer. Funct. Anal. Optim.}, 35 (2014), 1579.  doi: 10.1080/01630563.2014.908211.  Google Scholar

[49]

F. Martínez-Giménez, P. Oprocha and A. Peris, Distributional chaos for backward shifts,, \emph{J. Math. Anal. Appl.}, 351 (2009), 607.  doi: 10.1016/j.jmaa.2008.10.049.  Google Scholar

[50]

F. Martínez-Giménez, P. Oprocha and A. Peris, Distributional chaos for operators with full scrambled sets,, \emph{Math. Z.}, 274 (2013), 603.  doi: 10.1007/s00209-012-1087-8.  Google Scholar

[51]

Q. Menet, Linear chaos and frequent hypercyclicity,, \emph{Trans. Amer. Math. Soc.}, ().   Google Scholar

[52]

G. Metafune, $L_p$-spectrum of Ornstein-Uhlenbeck operators,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)}, 30 (2001), 97.   Google Scholar

[53]

V. Müller, On the Salas theorem and hypercyclicity of $f(T)$,, \emph{Integral Equations Operator Theory}, 67 (2010), 439.  doi: 10.1007/s00020-010-1791-x.  Google Scholar

[54]

P. Oprocha, A quantum harmonic oscillator and strong chaos,, \emph{J. Phys. A}, 39 (2006), 14559.  doi: 10.1088/0305-4470/39/47/003.  Google Scholar

[55]

B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval,, \emph{Trans. Amer. Math. Soc.}, 344 (1994), 737.  doi: 10.2307/2154504.  Google Scholar

[56]

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