Densely branching trees as models for H\'enon-like and Lozi-like attractors

TL;DR

This paper demonstrates that Hénon-like and Lozi-like attractors can be modeled by shift homeomorphisms on inverse limits of metric trees with dense branch points, linking topology and dynamics in these complex systems.

Contribution

It introduces a novel 1-dimensional tree-based model for Hénon-like and Lozi-like attractors, providing the first such models for these two-parameter families in the plane.

Findings

Hénon-like and Lozi-like attractors are conjugate to shift homeomorphisms on inverse limits of metric trees.

The models approximate the topology of attractors and accurately reflect their dynamics.

No simpler 1-dimensional models can represent these attractors.

Abstract

Inspired by a recent work of Crovisier and Pujals on mildly dissipative diffeomorphisms of the plane, we show that H\'enon-like and Lozi-like maps on their strange attractors are conjugate to natural extensions (a.k.a. shift homeomorphisms on inverse limits) of maps on metric trees with dense set of branch points. In consequence, these trees very well approximate the topology of the attractors, and the maps on them give good models of the dynamics. To the best of our knowledge, these are the first examples of canonical two-parameter families of attractors in the plane for which one is guaranteed such a 1-dimensional locally connected model tying together topology and dynamics of these attractors. For the H\'enon maps this applies to a positive Lebesgue measure parameter set generalizing the Benedicks-Carleson parameters, the Wang-Young parameter set, and sheds more light onto the result of Barge from 1987, who showed that there exist parameter values for which H\'enon maps on their attractors are not natural extensions of any maps on branched 1-manifolds. For the Lozi maps the result applies to an open set of parameters given by Misiurewicz in 1980. Our result can be seen as a generalization to the non-uniformly hyperbolic world of a classical result of Williams from 1967. We also show that no simpler 1-dimensional models exist.