Backward Julia Sets for a Class of p-adic Hénon-like Maps

Source abstract

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