Attractors associated to a family of hyperbolic $p$-adic plane automorphisms

TL;DR

This paper studies a family of hyperbolic automorphisms over non-Archimedean fields, revealing chaotic attractors conjugate to shift maps and calculating their non-integer Hausdorff dimension.

Contribution

It introduces a new class of hyperbolic automorphisms with explicit chaotic attractors and measures in the non-Archimedean setting, including their Hausdorff dimension.

Findings

Existence of chaotic attractors conjugate to shift maps

Support of a natural invariant measure describing orbit distribution

Hausdorff dimension of attractors is non-integer

Abstract

We consider a certain two-parameter family of automorphisms of the affine plane over a complete, locally compact non-Archimedean field. Each of these automorphisms admits a chaotic attractor on which it is topologically conjugate to a full two-sided shift map, and the attractor supports a unit Borel measure which describes the distribution of the forward orbit of Haar-almost all points in the basin of attraction. We also compute the Hausdorff dimension of the attractor, which is non-integral.