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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

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 & 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.  Google Scholar

[2]

X. Barrachina and A. Peris, Distributionally chaotic translation semigroups, J. Difference Equ. Appl., 18 (2012), 751-761. doi: 10.1080/10236198.2011.625945.  Google Scholar

[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.  Google Scholar

[4]

F. Bayart and É. Matheron, "Dynamics of Linear Operators,'' $1^{st}$ edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511581113.  Google Scholar

[5]

B. Beauzamy, "Introduction to Operator Theory and Invariant Subspaces,'' $1^{st}$ edition, North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[8]

N.C. Bernardes, A. Bonilla, V. Müller and A. Peris, Distributional chaos for linear operators,, preprint., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

K.J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations,'' $1^{st}$ edition, Springer-Verlag, New York, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

B. Hou, G. Tian and L. Shi, Some dynamical properties for linear operators, Illinois J. Math., 53 (2009), 857-864.  Google Scholar

[19]

H. König, On the Fourier-coefficients of vector-valued functions, Math. Nachr., 152 (1991), 215-227. doi: 10.1002/mana.19911520118.  Google Scholar

[20]

E. M. Mangino and A. Peris, Frequently hypercyclic semigroups, Studia Math., 202 (2011), 227-242. doi: 10.4064/sm202-3-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

G. Metafune, $L^p$-spectrum of Ornstein-Uhlenbeck operators, {Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 30 (2001), 97-124.  Google Scholar

[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.  Google Scholar

[25]

P. Oprocha, Distributional chaos revisited, Trans. Amer. Math. Soc., 361 (2009), 4901-4925. doi: 10.1090/S0002-9947-09-04810-7.  Google Scholar

[26]

P. Oprocha., Coherent lists and chaotic sets, Discrete Contin. Dyn. Syst., 31 (2011), 797-825. doi: 10.3934/dcds.2011.31.797.  Google Scholar

[27]

T. Ransford, Eigenvalues and power growth, Israel J. Math., 146 (2005), 93-110. doi: 10.1007/BF02773528.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

X. Barrachina and A. Peris, Distributionally chaotic translation semigroups, J. Difference Equ. Appl., 18 (2012), 751-761. doi: 10.1080/10236198.2011.625945.  Google Scholar

[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.  Google Scholar

[4]

F. Bayart and É. Matheron, "Dynamics of Linear Operators,'' $1^{st}$ edition, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511581113.  Google Scholar

[5]

B. Beauzamy, "Introduction to Operator Theory and Invariant Subspaces,'' $1^{st}$ edition, North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

N.C. Bernardes, A. Bonilla, V. Müller and A. Peris, Distributional chaos for linear operators,, preprint., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

K.J. Engel and R. Nagel, "One-parameter Semigroups for Linear Evolution Equations,'' $1^{st}$ edition, Springer-Verlag, New York, 2000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

B. Hou, G. Tian and L. Shi, Some dynamical properties for linear operators, Illinois J. Math., 53 (2009), 857-864.  Google Scholar

[19]

H. König, On the Fourier-coefficients of vector-valued functions, Math. Nachr., 152 (1991), 215-227. doi: 10.1002/mana.19911520118.  Google Scholar

[20]

E. M. Mangino and A. Peris, Frequently hypercyclic semigroups, Studia Math., 202 (2011), 227-242. doi: 10.4064/sm202-3-2.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

G. Metafune, $L^p$-spectrum of Ornstein-Uhlenbeck operators, {Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 30 (2001), 97-124.  Google Scholar

[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.  Google Scholar

[25]

P. Oprocha, Distributional chaos revisited, Trans. Amer. Math. Soc., 361 (2009), 4901-4925. doi: 10.1090/S0002-9947-09-04810-7.  Google Scholar

[26]

P. Oprocha., Coherent lists and chaotic sets, Discrete Contin. Dyn. Syst., 31 (2011), 797-825. doi: 10.3934/dcds.2011.31.797.  Google Scholar

[27]

T. Ransford, Eigenvalues and power growth, Israel J. Math., 146 (2005), 93-110. doi: 10.1007/BF02773528.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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