$p$-adic dynamical systems of $(3,1)$-rational functions with unique fixed point

Abstract

We describe the set of all (3, 1)-rational functions given on the set of complex p-adic field 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 Cp there is a point x = x(f ) ∈ 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 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.

Dynamics of a population with two equal dominated species

“…Problems of like this appear, for example, in electrical circuits that have switches, mechanical devices in which components impact with each other or have free play, problems with friction, sliding or squealing, many control systems and models in the social and financial sciences where continuous change can trigger discrete actions [2]. See also motivating examples of a piecewise-smooth systems: generated by the floor function ( [19], [22]) and p-adic dynamical systems (see [1], [13], [18], [20]).…”