Abstract
We consider a family of (2, 1)-rational functions given on the set of p-adic field \(Q_p\). Each such function has a unique fixed point. We study ergodicity properties of the dynamical systems generated by (2, 1)-rational functions. For each such function we describe all possible invariant spheres. We characterize ergodicity of each p-adic dynamical system with respect to Haar measure reduced on each invariant sphere. In particular, we found an invariant spheres on which the dynamical system is ergodic and on all other invariant spheres the dynamical systems are not ergodic.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Albeverio, S., Rozikov, U.A., Sattarov, I.A.: p-adic (2,1)-rational dynamical systems. J. Math. Anal. Appl. 398(2), 553–566 (2013)
Albeverio, S., Khrennikov, A., Tirozzi, B., De Smedt, S.: \(p\)-adic dynamical systems. Theor. Math. Phys. 114, 276–287 (1998)
Gundlach, V.M., Khrennikov, A., Lindahl, K.O.: On ergodic behavior of \(p\)-adic dynamical systems. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4, 569–577 (2001)
Memić, N.: Characterization of ergodic rational functions on the set 2-adic units. Int. J. Number Theory 13, 1119–1128 (2017)
Mukhamedov, F.M., Rozikov, U.A.: On rational \(p\)-adic dynamical systems. Methods Funct. Anal. Topol. 10(2), 21–31 (2004)
Peitgen, H.-O., Jungers, H., Saupe, D.: Chaos Fractals. Springer, Heidelberg (1992)
Rozikov, U.A., Sattarov, I.A.: On a non-linear p-adic dynamical system. \(p\)-adic numbers, ultrametric. Anal. Appl. 6(1), 53–64 (2014)
Rozikov, U.A., Sattarov, I.A.: \(p\)-adic dynamical systems of (2,2)-rational functions with unique fixed point. Chaos, Solitons Fractals 105, 260–270 (2017)
Sattarov, I.A.: \(p\)-adic (3,2)-rational dynamical systems. \(p\)-Adic Numbers, Ultrametric. Anal. Appl. 7(1), 39–55 (2015)
Walters, P.: An Introduction to Ergodic Theory. Springer, Berlin (1982)
Acknowledgements
The author expresses his deep gratitude to U. Rozikov for setting up the problem and for the useful suggestions. He also thanks both referees for helpful comments. In particular, a suggestion of a referee was helpful to simplify the proof of Theorem 3.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Sattarov, I.A. (2018). Ergodicity Properties of p-Adic (2, 1)-Rational Dynamical Systems with Unique Fixed Point. In: Ibragimov, Z., Levenberg, N., Rozikov, U., Sadullaev, A. (eds) Algebra, Complex Analysis, and Pluripotential Theory. USUZCAMP 2017. Springer Proceedings in Mathematics & Statistics, vol 264. Springer, Cham. https://doi.org/10.1007/978-3-030-01144-4_18
Download citation
DOI: https://doi.org/10.1007/978-3-030-01144-4_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-01143-7
Online ISBN: 978-3-030-01144-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)