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Asymptotically periodic solutions of neutral partial differential equations with infinite delay
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 |
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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