Non-Archimedean H\'enon maps, attractors, and horseshoes

TL;DR

This paper investigates the dynamics of non-Archimedean Hénon maps over certain fields, characterizing their Julia sets, attractors, and symbolic dynamics, with numerical evidence of complex behaviors over various p-adic fields.

Contribution

It provides a detailed analysis of Julia sets and attractors for non-Archimedean Hénon maps, including conditions for their existence and properties, and establishes conjugacy to symbolic shifts in certain parameter regions.

Findings

Filled Julia sets can be empty, bounded, or equal to the unit ball.

Existence of infinite attractors supporting SRB-like measures over ${f Q}_3$.

Topological conjugacy to shift maps in specific parameter regions.

Abstract

We study the dynamics of the H\'enon map defined over complete, locally compact non-Archimedean fields of odd residue characteristic. We establish basic properties of its one-sided and two-sided filled Julia sets, and we determine, for each H\'enon map, whether these sets are empty or nonempty, whether they are bounded or unbounded, and whether they are equal to the unit ball or not. On a certain region of the parameter space we show that the filled Julia set is an attractor. We prove that, for infinitely many distinct H\'enon maps over ${\mathbb Q}_3$, this attractor is infinite and supports an SRB-type measure describing the distribution of all nearby forward orbits. We include some numerical calculations which suggest the existence of such infinite attractors over ${\mathbb Q}_5$ and ${\mathbb Q}_7$ as well. On a different region of the parameter space, we show that the H\'enon map is topologically conjugate on its filled Julia set to the two-sided shift map on the space of bisequences in two symbols.