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

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.