# American Institute of Mathematical Sciences

September  2013, 33(9): 4233-4237. doi: 10.3934/dcds.2013.33.4233

## Periodic points and periods for operators on hilbert space

 1 School of Mathematics and Statistics, University of Hyderabad, Hyderabad, India, India, India

Received  November 2010 Revised  February 2011 Published  March 2013

We characterize the sets of periodic points of bounded linear operators on a Hilbert space $H$. We also find the pairs $(A,M)$, where $A \subset \mathbb{N}$, $M \subset H$ such that there exists a bounded linear operator $T$ on $H$ with $A$ as the set of periods and $M$ as the set of periodic points.
Citation: P. Chiranjeevi, V. Kannan, Sharan Gopal. Periodic points and periods for operators on hilbert space. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4233-4237. doi: 10.3934/dcds.2013.33.4233
##### References:
 [1] K. Ali Akbar, V. Kannan, Sharan Gopal and P. Chiranjeevi, The set of periods of periodic points of a linear operator, Linear Algebra and its Applications, 431 (2009), 241-246. doi: 10.1016/j.laa.2009.02.027. [2] I. N. Baker, Fixpoints of polynomials and rational functions, J. London Math. Soc., 39 (1964), 615-622. [3] L. S. Block and W. A. Coppel, "Dynamics in One Dimension," Lecture Notes in Mathematics, Vol. 1513, Springer-Verlag, Berlin, 1992. [4] M. Brin and G. Stuck, "Introduction to Dynamical Systems," Cambridge University Press, 2002. doi: 10.1017/CBO9780511755316. [5] J. P. Delahaye, The set of periodic points, Amer. Math. Monthly, 88 (1981), 646-651. doi: 10.2307/2320668. [6] R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," Second edition, Addison-wesley Publishing Company Advanced Book Program, Redwood City, CA, 1989. [7] V. Kannan, P. V. S. P. Saradhi and S. P. Seshasai, A generalization of Sarkovskii's theorem to higher dimensions, Special Volume to Felicitate Prof. Dr. R. S. Mishra on the Occasion of his 80th Birthday, J. Nat. Acad. Math. India, 11 (1997), 69-82 (1999). [8] V. Kannan, I. Subramania Pillai, K. Ali Akbar and B. Sankararao, The set of periods of periodic points of a toral automorphism, Topology Proceedings, 37 (2011), 1-14. [9] W. Rudin, "Functional Analysis," Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. [10] Sesha Sai, "Symbolic Dynamics for Complete Classification," Ph.D. Thesis, University of Hyderabad, 2000. [11] T. K. Subrahmonian Moothathu, Set of periods of additive cellular automata, Theoretical Computer Science, 352 (2006), 226-231. doi: 10.1016/j.tcs.2005.10.050. [12] I. Subramania Pillai, K. Ali Akbar, V. Kannan and B. Sankararao, Sets of all periodic points of a toral automorphism, J. Math. Anal. Appl., 366 (2010), 367-371. doi: 10.1016/j.jmaa.2009.12.032.

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##### References:
 [1] K. Ali Akbar, V. Kannan, Sharan Gopal and P. Chiranjeevi, The set of periods of periodic points of a linear operator, Linear Algebra and its Applications, 431 (2009), 241-246. doi: 10.1016/j.laa.2009.02.027. [2] I. N. Baker, Fixpoints of polynomials and rational functions, J. London Math. Soc., 39 (1964), 615-622. [3] L. S. Block and W. A. Coppel, "Dynamics in One Dimension," Lecture Notes in Mathematics, Vol. 1513, Springer-Verlag, Berlin, 1992. [4] M. Brin and G. Stuck, "Introduction to Dynamical Systems," Cambridge University Press, 2002. doi: 10.1017/CBO9780511755316. [5] J. P. Delahaye, The set of periodic points, Amer. Math. Monthly, 88 (1981), 646-651. doi: 10.2307/2320668. [6] R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," Second edition, Addison-wesley Publishing Company Advanced Book Program, Redwood City, CA, 1989. [7] V. Kannan, P. V. S. P. Saradhi and S. P. Seshasai, A generalization of Sarkovskii's theorem to higher dimensions, Special Volume to Felicitate Prof. Dr. R. S. Mishra on the Occasion of his 80th Birthday, J. Nat. Acad. Math. India, 11 (1997), 69-82 (1999). [8] V. Kannan, I. Subramania Pillai, K. Ali Akbar and B. Sankararao, The set of periods of periodic points of a toral automorphism, Topology Proceedings, 37 (2011), 1-14. [9] W. Rudin, "Functional Analysis," Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. [10] Sesha Sai, "Symbolic Dynamics for Complete Classification," Ph.D. Thesis, University of Hyderabad, 2000. [11] T. K. Subrahmonian Moothathu, Set of periods of additive cellular automata, Theoretical Computer Science, 352 (2006), 226-231. doi: 10.1016/j.tcs.2005.10.050. [12] I. Subramania Pillai, K. Ali Akbar, V. Kannan and B. Sankararao, Sets of all periodic points of a toral automorphism, J. Math. Anal. Appl., 366 (2010), 367-371. doi: 10.1016/j.jmaa.2009.12.032.
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