Backward Julia sets for a class of p-adic Hénon like maps

Abstract

In this work we study the backward filled Julia sets of a class of $p$-adic polynomial maps $f:{\mathbb{Q}}_p^2⟶{\mathbb{Q}}_p^2$ defined by $f(x,y)=(xy+c,x)$, where $c\in {\mathbb{Q}}_p$ is a $p$-adic number. In particular, if $|c|\le 1$, 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.

20 pages, 3 figures