Forward Julia Sets for a Class of $$p$$-Adic Hénon Like Maps

In this work we study the forward 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, we prove that if $$|c|< 1$$ , then the forward filled Julia set has infinity Haar measure and contains the additive group of $$p$$ -adic integers. Furthermore, excepted for a bounded subset of the set of $$p$$ -adic integers, we prove that the orbit of all points of the filled Julia set converges to a fixed point of $$f$$ . On the other hand, if $$|c|>1$$ , then we exhibit a bounded set such that, the filled Julia set is characterized by the points whose the orbit enter in this set and it never leaves it after each iteration of the map $$f$$ . Moreover, for all parameter $$c$$ , we prove that both coordinates of the orbit of all points that are not in the filled Julia set goes to infinity in $$p$$ -adic norm.