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

Distributional chaos for strongly continuous semigroups of operators

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.
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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