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Inspired by a recent work of Crovisier and Pujals on mildly dissipativediffeomorphisms of the plane, we show that H\'enon-like and Lozi-like maps ontheir strange attractors are conjugate to natural extensions (a.k.a. shifthomeomorphisms on inverse limits) of maps on metric trees with dense set ofbranch points. In consequence, these trees very well approximate the topologyof the attractors, and the maps on them give good models of the dynamics. Tothe best of our knowledge, these are the first examples of canonicaltwo-parameter families of attractors in the plane for which one is guaranteedsuch a 1-dimensional locally connected model tying together topology anddynamics of these attractors. For the H\'enon maps this applies to a positiveLebesgue measure parameter set generalizing the Benedicks-Carleson parameters,the Wang-Young parameter set, and sheds more light onto the result of Bargefrom 1987, who showed that there exist parameter values for which H\'enon mapson their attractors are not natural extensions of any maps on branched1-manifolds. For the Lozi maps the result applies to an open set of parametersgiven by Misiurewicz in 1980. Our result can be seen as a generalization to thenon-uniformly hyperbolic world of a classical result of Williams from 1967. Wealso show that no simpler 1-dimensional models exist.

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