Distributional chaos for strongly continuous semigroups of operators

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September 2013, 12(5): 2069-2082. doi: 10.3934/cpaa.2013.12.2069

Distributional chaos for strongly continuous semigroups of operators

Angela A. Albanese ^1, , Xavier Barrachina ^2, , Elisabetta M. Mangino ^3, and Alfredo Peris ^4,

1.	Department of Mathematics “E. De Giorgi”, University of Salento, P.O. Box 193, Via Per Arnesano, 73100 Lecce, Italy
2.	Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, Edifici 8E, 46022 València, Spain
3.	Dipartimento di Matematica e Fisica "Ennio De Giorgi'', Università del Salento, Via Per Arnesano P.O. Box 193, 73100 Lecce, Italy
4.	Departament de Matemàtica Aplicada & IUMPA Edifici 7A, Universitat Politècnica de València, E-46022, València

Received February 2012 Revised October 2012 Published January 2013

Abstract
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Distributional chaos for strongly continuous semigroups is studied and characterized. It is shown to be equivalent to the existence of a distributionally irregular vector. Finally, a sufficient condition for distributional chaos on the point spectrum of the generator of the semigroup is presented. An application to the semigroup generated in $L^2(R)$ by a translation of the Ornstein-Uhlenbeck operator is also given.

Keywords: Distributional chaos, hypercyclic operators, irregular vectors, criterion for distributional chaos., $C_0$-semigroups.

Mathematics Subject Classification: 47A16, 47D06, 37D4.

Citation: 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

References:

[1]	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.
[2]	X. Barrachina and A. Peris, Distributionally chaotic translation semigroups, J. Difference Equ. Appl., 18 (2012), 751-761. doi: 10.1080/10236198.2011.625945.
[3]	F. Bayart and S. Grivaux, Hypercyclicity and unimodular point spectrum, J. Funct. Anal., 226 (2005), 281-300. doi: 10.1016/j.jfa.2005.06.001.
[4]	F. Bayart and É. Matheron, "Dynamics of Linear Operators,'' $1^{st}$ edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511581113.
[5]	B. Beauzamy, "Introduction to Operator Theory and Invariant Subspaces,'' $1^{st}$ edition, North-Holland Publishing Co., Amsterdam, 1988.
[6]	T. Bermúdez, A. Bonilla, J. A. Conejero and A. Peris, Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces, Studia Math., 170 (2005), 57-75. doi: 10.4064/sm170-1-3.
[7]	T. Bermúdez, A. Bonilla, F. Martínez-Giménez and A. Peris, Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl., 373 (2011), 83-93. doi: 10.1016/j.jmaa.2010.06.011.
[8]	N.C. Bernardes, A. Bonilla, V. Müller and A. Peris, Distributional chaos for linear operators,, preprint., ().
[9]	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.
[10]	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.
[11]	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.
[12]	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.
[13]	K.J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations,'' $1^{st}$ edition, Springer-Verlag, New York, 2000.
[14]	M. C. Gómez-Collado, F. Martínez-Giménez, A. Peris and F. Rodenas, Slow growth for universal harmonic functions, J. Inequal. Appl., 2010, Article ID 253690, (2010), 6 pp. doi: 10.1155/2010/253690.
[15]	S. Grivaux, A new class of frequently hypercyclic operators, Indiana Univ. Math. J., 60 (2011), 1177-1202. doi: 10.1512/iumj.2011.60.4350.
[16]	K. G. Grosse-Erdmann and A. Peris Manguillot, "Linear Chaos,'' $1^{st}$ edition, Universitext, Springer, London, 2011. doi: 10.1007/978-1-4471-2170-1.
[17]	B. Hou, P. Cui and Y. Cao, Chaos for Cowen-Douglas operators, Proc. Amer. Math. Soc., 138 (2010), 929-936. doi: 10.1090/S0002-9939-09-10046-1.
[18]	B. Hou, G. Tian and L. Shi, Some dynamical properties for linear operators, Illinois J. Math., 53 (2009), 857-864.
[19]	H. König, On the Fourier-coefficients of vector-valued functions, Math. Nachr., 152 (1991), 215-227. doi: 10.1002/mana.19911520118.
[20]	E. M. Mangino and A. Peris, Frequently hypercyclic semigroups, Studia Math., 202 (2011), 227-242. doi: 10.4064/sm202-3-2.
[21]	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.
[22]	F. Martínez-Giménez, P. Oprocha, A. Peris, Distributional chaos for operators with full scrambled sets,, {Math. Z.} (To appear)., (). doi: 10.1007/s00209-012-1087-8.
[23]	G. Metafune, $L^p$-spectrum of Ornstein-Uhlenbeck operators, {Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 30 (2001), 97-124.
[24]	P. Oprocha, A quantum harmonic oscillator and strong chaos, J. Phys. A, 39 (2006), 14559-14565. doi: 10.1088/0305-4470/39/47/003.
[25]	P. Oprocha, Distributional chaos revisited, Trans. Amer. Math. Soc., 361 (2009), 4901-4925. doi: 10.1090/S0002-9947-09-04810-7.
[26]	P. Oprocha., Coherent lists and chaotic sets, Discrete Contin. Dyn. Syst., 31 (2011), 797-825. doi: 10.3934/dcds.2011.31.797.
[27]	T. Ransford, Eigenvalues and power growth, Israel J. Math., 146 (2005), 93-110. doi: 10.1007/BF02773528.
[28]	R. Rudnicki, 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.
[29]	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.
[30]	X. Wu and P. Zhu, The principal measure of a quantum harmonic oscillator, J. Phys. A, 44, 505101 (2011), 6 pp. doi: 10.1088/1751-8113/44/50/505101.

show all references

References:

[1]	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.
[2]	X. Barrachina and A. Peris, Distributionally chaotic translation semigroups, J. Difference Equ. Appl., 18 (2012), 751-761. doi: 10.1080/10236198.2011.625945.
[3]	F. Bayart and S. Grivaux, Hypercyclicity and unimodular point spectrum, J. Funct. Anal., 226 (2005), 281-300. doi: 10.1016/j.jfa.2005.06.001.
[4]	F. Bayart and É. Matheron, "Dynamics of Linear Operators,'' $1^{st}$ edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511581113.
[5]	B. Beauzamy, "Introduction to Operator Theory and Invariant Subspaces,'' $1^{st}$ edition, North-Holland Publishing Co., Amsterdam, 1988.
[6]	T. Bermúdez, A. Bonilla, J. A. Conejero and A. Peris, Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces, Studia Math., 170 (2005), 57-75. doi: 10.4064/sm170-1-3.
[7]	T. Bermúdez, A. Bonilla, F. Martínez-Giménez and A. Peris, Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl., 373 (2011), 83-93. doi: 10.1016/j.jmaa.2010.06.011.
[8]	N.C. Bernardes, A. Bonilla, V. Müller and A. Peris, Distributional chaos for linear operators,, preprint., ().
[9]	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.
[10]	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.
[11]	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.
[12]	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.
[13]	K.J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations,'' $1^{st}$ edition, Springer-Verlag, New York, 2000.
[14]	M. C. Gómez-Collado, F. Martínez-Giménez, A. Peris and F. Rodenas, Slow growth for universal harmonic functions, J. Inequal. Appl., 2010, Article ID 253690, (2010), 6 pp. doi: 10.1155/2010/253690.
[15]	S. Grivaux, A new class of frequently hypercyclic operators, Indiana Univ. Math. J., 60 (2011), 1177-1202. doi: 10.1512/iumj.2011.60.4350.
[16]	K. G. Grosse-Erdmann and A. Peris Manguillot, "Linear Chaos,'' $1^{st}$ edition, Universitext, Springer, London, 2011. doi: 10.1007/978-1-4471-2170-1.
[17]	B. Hou, P. Cui and Y. Cao, Chaos for Cowen-Douglas operators, Proc. Amer. Math. Soc., 138 (2010), 929-936. doi: 10.1090/S0002-9939-09-10046-1.
[18]	B. Hou, G. Tian and L. Shi, Some dynamical properties for linear operators, Illinois J. Math., 53 (2009), 857-864.
[19]	H. König, On the Fourier-coefficients of vector-valued functions, Math. Nachr., 152 (1991), 215-227. doi: 10.1002/mana.19911520118.
[20]	E. M. Mangino and A. Peris, Frequently hypercyclic semigroups, Studia Math., 202 (2011), 227-242. doi: 10.4064/sm202-3-2.
[21]	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.
[22]	F. Martínez-Giménez, P. Oprocha, A. Peris, Distributional chaos for operators with full scrambled sets,, {Math. Z.} (To appear)., (). doi: 10.1007/s00209-012-1087-8.
[23]	G. Metafune, $L^p$-spectrum of Ornstein-Uhlenbeck operators, {Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 30 (2001), 97-124.
[24]	P. Oprocha, A quantum harmonic oscillator and strong chaos, J. Phys. A, 39 (2006), 14559-14565. doi: 10.1088/0305-4470/39/47/003.
[25]	P. Oprocha, Distributional chaos revisited, Trans. Amer. Math. Soc., 361 (2009), 4901-4925. doi: 10.1090/S0002-9947-09-04810-7.
[26]	P. Oprocha., Coherent lists and chaotic sets, Discrete Contin. Dyn. Syst., 31 (2011), 797-825. doi: 10.3934/dcds.2011.31.797.
[27]	T. Ransford, Eigenvalues and power growth, Israel J. Math., 146 (2005), 93-110. doi: 10.1007/BF02773528.
[28]	R. Rudnicki, 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.
[29]	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.
[30]	X. Wu and P. Zhu, The principal measure of a quantum harmonic oscillator, J. Phys. A, 44, 505101 (2011), 6 pp. doi: 10.1088/1751-8113/44/50/505101.

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