Ergodicity Properties of p-Adic (2, 1)-Rational Dynamical Systems with Unique Fixed Point
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.
Keywordsp-adic numbers Rational function Dynamical system Ergodic
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.