2018/07/27 by Alexandro Luna, Luna, A. R., U. A. Rozikov +3
Computer Science · Mathematics · #11S99 #12J12 #46S10 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1807.11561
openalex publication_date 2018/07/27 · openalex created_date 2019/03/02 · openalex updated_date 2026/07/28
We describe the set of all (3,1)-rational functions given on the set of complex p-adic field \mathbb Cp and having a unique fixed point. We study p-adic dynamical systems generated by such (3,1)-rational functions and show that the fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. We obtain Siegel disks of these dynamical systems. Moreover an upper bound for the set of limit points of each trajectory is given. For each (3,1)-rational function on \mathbb Cp there is a point x= x(f)∈ \mathbb Cp which is zero in its denominator. We give explicit formulas of radii of spheres (with the center at the fixed point) containing some points that the trajectories (under actions of f) of the points after a finite step come to x. For a class of (3,1)-rational functions defined on the set of p-adic numbers \mathbb Qp we study ergodicity properties of the corresponding dynamical systems. We show that if p≥ 3 then the p-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure. For p=2, under some conditions we prove non ergodicity and show that there exists a sphere on which the dynamical system is ergodic. Finally, we give a characterization of periodic orbits and some uniformly local properties of the (3.1)-rational functions.