$p$-adic $(2,1)$-rational dynamical systems

TL;DR

This paper analyzes the dynamics of $(2,1)$-rational functions over $p$-adic fields, revealing fixed points, cycles, invariant spheres, and basins of attraction, with detailed classifications based on fixed point types and parameter variations.

Contribution

It provides a comprehensive classification of $p$-adic $(2,1)$-rational dynamical systems, including fixed points, cycles, Siegel disks, and invariant spheres, which was not previously detailed.

Findings

Existence of 2-periodic cycles and their attraction properties.

Characterization of basins of attraction for different fixed point types.

Identification of conditions for invariant spheres and Siegel disks.

Abstract

We investigate the trajectory of an arbitrary $(2,1)$-rational $p$-adic dynamical system in a complex $p$-adic field $\C_p$. (i) In the case where there is no fixed point we show that the $p$-adic dynamical system has a 2-periodic cycle $x_1, x_2$. If it is attracting then it attracts each trajectory which starts from an element of a ball of radius $r=|x_1-x_2|_p$ with the center at $x_1$ or at $x_2$. If the 2-periodic cycle is an indifferent, then in each step the balls transfer to each other. All the other spheres with radius $>r$ and the center at $x_1$ and $x_2$ are invariant independently of the attractiveness of the cycle. (ii) In the case where the fixed point $x_0$ is unique we prove that if the point is attracting then there exists $\delta>0$, such that the basin of attraction for $x_0$ is the ball of radius $\delta$ and the center at $x_0$ and any sphere with radius $\geq \delta$ is invariant. If $x_0$ is an indifferent point then all spheres with the center at $x_0$ are invariant. If $x_0$ is a repelling point then there exits $\delta>0$, such that the trajectory which starts at an element of the ball of radius $\delta$ with the center in $x_0$ leaves this ball, whereas any sphere with radius $\geq \delta$ is invariant. (iii) In case of existence of two fixed points, we show that Siegel disks may either coincide or be disjoint for different fixed points of the dynamical system. Besides, we find the basin of the attractor of the system. Varying the parameters it is proven that there exists an integer $k\geq 2$, and spheres $S_{r_1}(x_i), ..., S_{r_k}(x_i)$ such that the limiting trajectory will be periodically traveling the spheres $S_{r_j}$. For some values of the parameters there are trajectories which go arbitrary far from the fixed points.