# 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 & Continuous Dynamical Systems - A, 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.  doi: 10.1016/j.laa.2009.02.027.  Google Scholar [2] I. N. Baker, Fixpoints of polynomials and rational functions,, J. London Math. Soc., 39 (1964), 615.   Google Scholar [3] L. S. Block and W. A. Coppel, "Dynamics in One Dimension,", Lecture Notes in Mathematics, (1513).   Google Scholar [4] M. Brin and G. Stuck, "Introduction to Dynamical Systems,", Cambridge University Press, (2002).  doi: 10.1017/CBO9780511755316.  Google Scholar [5] J. P. Delahaye, The set of periodic points,, Amer. Math. Monthly, 88 (1981), 646.  doi: 10.2307/2320668.  Google Scholar [6] R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,", Second edition, (1989).   Google Scholar [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, 11 (1997), 69.   Google Scholar [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.   Google Scholar [9] W. Rudin, "Functional Analysis,", Second edition, (1991).   Google Scholar [10] Sesha Sai, "Symbolic Dynamics for Complete Classification,", Ph.D. Thesis, (2000).   Google Scholar [11] T. K. Subrahmonian Moothathu, Set of periods of additive cellular automata,, Theoretical Computer Science, 352 (2006), 226.  doi: 10.1016/j.tcs.2005.10.050.  Google Scholar [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.  doi: 10.1016/j.jmaa.2009.12.032.  Google Scholar

show all references

##### 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.  doi: 10.1016/j.laa.2009.02.027.  Google Scholar [2] I. N. Baker, Fixpoints of polynomials and rational functions,, J. London Math. Soc., 39 (1964), 615.   Google Scholar [3] L. S. Block and W. A. Coppel, "Dynamics in One Dimension,", Lecture Notes in Mathematics, (1513).   Google Scholar [4] M. Brin and G. Stuck, "Introduction to Dynamical Systems,", Cambridge University Press, (2002).  doi: 10.1017/CBO9780511755316.  Google Scholar [5] J. P. Delahaye, The set of periodic points,, Amer. Math. Monthly, 88 (1981), 646.  doi: 10.2307/2320668.  Google Scholar [6] R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,", Second edition, (1989).   Google Scholar [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, 11 (1997), 69.   Google Scholar [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.   Google Scholar [9] W. Rudin, "Functional Analysis,", Second edition, (1991).   Google Scholar [10] Sesha Sai, "Symbolic Dynamics for Complete Classification,", Ph.D. Thesis, (2000).   Google Scholar [11] T. K. Subrahmonian Moothathu, Set of periods of additive cellular automata,, Theoretical Computer Science, 352 (2006), 226.  doi: 10.1016/j.tcs.2005.10.050.  Google Scholar [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.  doi: 10.1016/j.jmaa.2009.12.032.  Google Scholar
 [1] Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463 [2] Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266 [3] Jiahao Qiu, Jianjie Zhao. Maximal factors of order $d$ of dynamical cubespaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278 [4] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [5] Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 [6] Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240 [7] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [8] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [9] Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055 [10] Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302 [11] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 [12] Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353 [13] Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355 [14] Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266 [15] Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168 [16] Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301 [17] Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291 [18] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321 [19] Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045 [20] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

2019 Impact Factor: 1.338