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Chaos for Cowen-Douglas operators
Author(s):
Bingzhe
Hou;
Puyu
Cui;
Yang
Cao
Journal:
Proc. Amer. Math. Soc.
138
(2010),
929-936.
MSC (2000):
Primary 47B37, 47B99;
Secondary 54H20, 37B99
Posted:
October 28, 2009
MathSciNet review:
2566559
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Abstract:
In this article, we provide a sufficient condition which gives Devaney chaos and distributional chaos for Cowen-Douglas operators. In fact, we obtain a distributionally chaotic criterion for bounded linear operators on Banach spaces.
References:
-
- 1.
- J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, Vol. 99, 1992, 332-334. MR 1157223 (93d:54059)
- 2.
- M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta Mathematica, Vol. 141(1), 1978, 187-261. MR 501368 (80f:47012)
- 3.
- R. L. Devaney, An introduction to chaotic dynamical systems, Benjamin/Cummings, 1986. MR 811850 (87e:58142)
- 4.
- J. Duan, X. C. Fu, P. D. Liu and A. Manning, A linear chaotic quantum harmonic oscillator, Appl. Math. Lett. 12(1), 1999, 15-19. MR 1663405 (99h:81054)
- 5.
- X. C. Fu and J. Duan, Infinite-dimensional linear dynamical systems with chaoticity, J. Nonlinear Sci. 9(2), 1999, 197-211. MR 1674380 (2000a:37003)
- 6.
- F. Martınez-Giménez, P. Oprocha and A. Peris, Distributional chaos for backward shifts, Journal of Mathematical Analysis and Applications, Vol. 351(2), 2009, 607-615. MR 2473967
- 7.
- K. G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math., Vol. 139(1), 2000, 47-68. MR 1763044 (2001f:47051)
- 8.
- W. Huang and X. Ye, Devaney chaos or 2-scattering implies Li-Yorke chaos, Topology Appl., Vol. 117, 2002, 259-272. MR 1874089 (2003b:37017)
- 9.
- T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82(10), 1975, 985-992. MR 0385028 (52:5898)
- 10.
- Gongfu Liao, Lidong Wang and Xiaodong Duan, A chaotic function with a distributively scrambled set of full Lebesgue measure, Nonlinear Analysis: Theory, Methods and Applications, Vol. 66(10), 2007, 2274-2280. MR 2311031 (2008c:37020)
- 11.
- Gongfu Liao, Zhenyan Chu and Qinjie Fan, Relations between mixing and distributional chaos, Chaos, Solitons and Fractals, in press, corrected proof, available online 16 September 2008.
- 12.
- J. Mai, Devaney chaos implies existence of
 -scrambled sets, Proc. Amer. Math. Soc. 132(9), 2004, 2761-2767. MR 2054803 (2005a:37022) - 13.
- P. Oprocha, A quantum harmonic oscillator and strong chaos, J. Phys. A 39(47), 2006, 14559-14565. MR 2277068 (2007i:81099)
- 14.
- B. Schweizer and J. Smıtal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344(2), 1994, 737-754. MR 1227094 (94k:58091)
- 15.
- S. Silverman, On maps with dense orbits and the definition of chaos, Rocky Mountain Jour. Math., Vol. 22, 1992, 353-375. MR 1159963 (93j:54033)
- 16.
- J. Smıtal and M. Štefánková, Distributional chaos for triangular maps, Chaos, Solitons and Fractals, Volume 21, Issue 5, 2004, 1125-1128. MR 2047330 (2005a:37017)
- 17.
- F. Balibrea, J. Smıtal and M. Štefánková, The three versions of distributional chaos, Chaos, Solitons and Fractals, Volume 23, Issue 5, 2005, 1581-1583. MR 2101573 (2005f:37040)
- 18.
- Hui Wang, Gongfu Liao and Qinjie Fan, Substitution systems and the three versions of distributional chaos, Topology and its Applications, Volume 156, Issue 2, 2008, 262-267. MR 2475113
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Additional Information:
Bingzhe
Hou
Affiliation:
Department of Mathematics, Jilin University, 130012, Changchun, People's Republic of China
Email:
houbz@jlu.edu.cn
Puyu
Cui
Affiliation:
Department of Mathematics, Dalian University of Technology, 116024, Dalian, People's Republic of China
Email:
cuipuyu1234@sina.com.cn
Yang
Cao
Affiliation:
Department of Mathematics, Jilin University, 130012, Changchun, People's Republic of China
Email:
caoyang@jlu.edu.cn
DOI:
10.1090/S0002-9939-09-10046-1
PII:
S 0002-9939(09)10046-1
Keywords:
Cowen-Douglas operators,
strongly mixing,
Devaney chaos,
distributional chaos.
Received by editor(s):
January 4, 2009,
Received by editor(s) in revised form:
April 16, 2009
Posted:
October 28, 2009
Additional Notes:
The first author is supported by the Youth Foundation of the Department of Mathematics, Jilin University
Communicated by:
Bryna Kra
Copyright of article:
Copyright
2009,
American Mathematical Society
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