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

TL;DR

This work analyzes backward Julia sets for the p-adic Hénon-like map $f(x,y)=(xy+c,x)$ on $\mathbb{Q}_p^2$ by establishing a partition-based framework that separates regions contributing to $\mathcal{K}^-$ from those that do not. It proves that $\mathcal{K}^-$ is bounded in $\mathbb{Z}_p^2$ when $|c|_p\le1$, and unbounded with infinite Haar measure when $|c|_p>1$, with a bounded trap set $\mathsf{C}_0\cup\mathsf{J}_0$ capturing backward orbits eventually. The proofs rely on detailed $p$-adic partitioning of the phase space and analyze border cases such as $|c|_p=1$. The work also raises questions about the structure of the full filled Julia set and potential symbolic dynamics equivalents, offering a foundation for further study of $p$-adic dynamical systems and invariant measures.

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.

Figures (3)

• Figure 1: Representation of the sets $\mathsf{F}$, $\mathsf{G}$, $\mathsf{H}$, $\mathsf{J}_i$ and $\mathsf{M}_i$ in $p$-adic norm, for $|c|>1$ and $i=1,2,3,4$.
• Figure 2: Representation of $\mathsf{J}_i$ and $\mathsf{M}_i$ in $p$-adic norm, for $|c|>1$.
• Figure 3: The partition $\mathsf{Q}= \mathsf{C}_0 \cup \mathsf{F} \cup \mathsf{G} \cup \mathsf{H} \cup \mathsf{M}$ in $p$-adic norm, for $|c|=1$.

Theorems & Definitions (35)

• Remark 1
• Theorem 3.1
• Remark 2
• Theorem 3.2
• Remark 3
• Theorem 3.3
• Remark 4
• Lemma 4.1
• proof
• Remark 5