# Ergodicity Properties of p-Adic (2, 1)-Rational Dynamical Systems with Unique Fixed Point

• Iskandar A. Sattarov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 264)

## 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.

## Keywords

p-adic numbers Rational function Dynamical system Ergodic

## Notes

### 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.

## References

1. 1.
Albeverio, S., Rozikov, U.A., Sattarov, I.A.: p-adic (2,1)-rational dynamical systems. J. Math. Anal. Appl. 398(2), 553–566 (2013)
2. 2.
Albeverio, S., Khrennikov, A., Tirozzi, B., De Smedt, S.: $$p$$-adic dynamical systems. Theor. Math. Phys. 114, 276–287 (1998)
3. 3.
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)
4. 4.
Memić, N.: Characterization of ergodic rational functions on the set 2-adic units. Int. J. Number Theory 13, 1119–1128 (2017)
5. 5.
Mukhamedov, F.M., Rozikov, U.A.: On rational $$p$$-adic dynamical systems. Methods Funct. Anal. Topol. 10(2), 21–31 (2004)
6. 6.
Peitgen, H.-O., Jungers, H., Saupe, D.: Chaos Fractals. Springer, Heidelberg (1992)
7. 7.
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)
8. 8.
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)
9. 9.
Sattarov, I.A.: $$p$$-adic (3,2)-rational dynamical systems. $$p$$-Adic Numbers, Ultrametric. Anal. Appl. 7(1), 39–55 (2015)
10. 10.
Walters, P.: An Introduction to Ergodic Theory. Springer, Berlin (1982)