$p$-adic dynamical systems of the function $a x^{-2}$

TL;DR

This paper investigates the behavior of $p$-adic dynamical systems generated by the function $a/x^2$, providing explicit formulas for iterations and analyzing fixed points, periodic points, and attraction basins in the complex $p$-adic setting.

Contribution

It introduces explicit formulas for iterates of the function and characterizes fixed points, periodic points, and basins of attraction depending on parameters $p$ and $a$.

Findings

Explicit formula for $n$-fold composition of $f(x)=a/x^2$.

Classification of fixed and periodic points based on parameters.

Description of basins of attraction and Siegel disks.

Abstract

In this paper we study $p$-adic dynamical systems generated by the function $f(x)={a\over x^2}$ in the set of complex $p$-adic numbers. We find an explicit formula for the $n$-fold composition of $f$ for any $n\geq 1$. Using this formula we give fixed points, periodic points, basin of attraction and Siegel disk of each fixed (periodic) point depending on parameters $p$ and $a$.