Backward Julia sets for a class of p-adic H\'enon like maps

TL;DR

This paper investigates the structure of backward filled Julia sets for a class of p-adic polynomial maps, revealing boundedness when |c| ≤ 1 and unboundedness with infinite measure when |c| > 1.

Contribution

It characterizes the boundedness and measure of backward Julia sets for p-adic Henon-like maps based on the parameter c.

Findings

Backward Julia sets are bounded in Z_p^2 when |c| ≤ 1.

Backward Julia sets are unbounded with infinite Haar measure when |c| > 1.

The results provide a clear dichotomy based on the p-adic norm of c.

Abstract

In this work we study the backward filled Julia sets of a class of $p$-adic polynomial maps $f:\mathbb{Q}_p^2\longrightarrow \mathbb{Q}_p^2$ defined by $f(x,y)=(xy+c,x)$, where $c\in\mathbb{Q}_p$ is a $p$-adic number. In particular, if $|c|\leq 1$, then we proved that the backward filled Julia set of $f$ is a bounded subset in $\mathbb{Z}_p^2$. On the other hand, if $|c|> 1$, then we prove that the backward filled Julia set of $f$ is an unbounded set and has infinity Haar measure.