Abstract
We show that any (1, 2)-rational function with a unique fixed point is topologically conjugate to a (2, 2)-rational function or to the function f(x) = ax/x2 + a. The case (2, 2) was studied in our previous paper, here we study the dynamical systems generated by the function f on the set of complex p-adic field ℂp. We show that the unique fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. We construct the corresponding Siegel disk of these dynamical systems. We determine a sufficiently small set containing the set of limit points. It is given all possible invariant spheres.We show that the p-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure on the set of p-adic numbers ℚp.Moreover some periodic orbits of the system are investigated.
Similar content being viewed by others
References
S. Albeverio, U. A. Rozikov and I. A. Sattarov, “p-Adic (2, 1)-rational dynamical systems,” J. Math. Anal. Appl. 398 (2), 553–566 (2013).
V. Anashin and A. Khrennikov, Applied Algebraic Dynamics, de Gruyter Expositions in Math. 49 (Walter de Gruyter, Berlin-New York, 2009).
N. Koblitz, p-Adic Numbers, p-Adic Analysis and Zeta-Function (Springer, Berlin, 1977).
H.-O. Peitgen, H. Jungers and D. Saupe, Chaos Fractals (Springer, Heidelberg-New York, 1992).
A. C. M. van Rooij, Non-Archimedean Functional Analysis, Monographs and Textbooks in Pure and AppliedMath. 51 (Marcel Dekker, Inc., New York, 1978).
U. A. Rozikov and I. A. Sattarov, “On a non-linear p-adic dynamical system,” p-Adic Numbers Ultrametric Anal. Appl. 6 (1), 53–64 (2014).
U. A. Rozikov and I. A. Sattarov, “p-Adic dynamical systems of (2, 2)-rational functions with unique fixed point,” Chaos Solit. Fract. 105, 260–270 (2017).
I. A. Sattarov, “p-Adic (3, 2)-rational dynamical systems,” p-Adic Numbers Ultrametric Anal. Appl. 7 (1), 39–55 (2015).
W. H. Schikhof, Ultrametric Calculus: An introduction to p-adic analysis, Cambridge Studies in Advanced Math. 4 (Cambridge Univ. Press, Cambridge, 2006).
V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, River Edge, N. Y., 1994).
P. Walters, An Introduction to Ergodic Theory (Springer, Berlin-Heidelberg-New York, 1982).
Author information
Authors and Affiliations
Corresponding author
Additional information
The text was submitted by the authors in English.
Rights and permissions
About this article
Cite this article
Rozikov, U.A., Sattarov, I.A. & Yam, S. p-Adic Dynamical Systems of the Function ax/x2 + a. P-Adic Num Ultrametr Anal Appl 11, 77–87 (2019). https://doi.org/10.1134/S2070046619010059
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070046619010059