Densely branching trees as models for Hénon-like and Lozi-like attractors

Abstract

Inspired by a recent work of Crovisier and Pujals on mildly dissipative diffeomorphisms of the plane, we show that Hénon-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 Hénon maps this applies to Benedicks-Carleson positive Lebesgue measure parameter set, and sheds more light onto the result of Barge from 1987, who showed that there exist parameter values for which Hénon maps on their attractors are not natural extensions of any maps on branched 1-manifolds. For Lozi maps the result applies to an open set of parameters given by Misiurewicz in 1980. In general, under mild dissipation, conjugacy holds if the attractor does not have an arc in common with a stable manifold, and the density of branch points holds when the attractor is a homoclinic class. Our result can be seen as a generalization to the non-uniformly hyperbolic world of a classical result of Williams from 1967.

On conjugacy of natural extensions of one-dimensional maps

“…In this case, there is an interval L ⊂ f (I i−1 ) such that x ∈ int(L) and L ∩ f ([u i , v i ]) = ∅. It follows from Proposition 4.2 that there is an interval I ⊂ I i−1 such that f (I ) = L. Observe that this choice of I satisfies condition (10)…”

“…In fact, it is known that for certain Hénon maps, this is never true [3]. Furthermore, Hénon and Lozi attractors discussed in [10] always contain non-degenerate arc components (such as branches of unstable manifolds), whereas the pseudo-arc contains no non-degenerate arcs at all. Moreover, as we have already mentioned, the interval maps f such that lim ← − ([0, 1], f ) is the pseudo-arc, and their natural extensions σ f have infinite topological entropy, but the Hénon and Lozi maps have finite entropy, bounded above by log 2.…”

“…Our paper is motivated by a recent result of the first and last author [10], in which it has been shown that for a class of mildly dissipative plane homeomorphisms that contains positive Lebesgue measure subsets of Lozi and Hénon maps, the dynamics on their attractors is conjugate to natural extensions of densely branching dendrite maps. In that context, the question arose whether these homeomorphisms could be also conjugate to natural extensions of maps on some simpler one-dimensional spaces, such as the interval [0, 1].…”

“…These examples, however, do not provide any new pieces of information for Hénon or Lozi maps. Moreover, it seems rather implausible that, for parameter values considered in [10], these maps would semi-conjugate to interval maps with a small folds property, should any of them semi-conjugate to any interval map at all. In fact, it is known that for certain Hénon maps, this is never true [3].…”

Critical points in higher dimensions, I: Reverse order of periodic orbit creations in the Lozi family

“…In one dimension, it is possible to classify maps by a finite-parameter family [Guc79,MT88]. In two dimensions, a finite-parameter family is not enough to solve the classification problem of a class of maps [HMT17,CP18,BŠ21].…”