In my talk I shall discuss my recent work with S. Stimac on
parametric families of mildly dissipative homeomorphisms on surfaces
[5]. Our study is motivated by recent novel approach of Crovisier and
Pujals [1], in which they introduced the notion of Strongly (Mildly)
Dissipative Diffeomorphisms (see also [2]). These maps are shown in
[1] to be very close in a certain sense to 1-dimensional maps. Within
the Misiurewicz parameter set [4] we construct a related reduction of
Lozi family [3] to maps on metric trees with dense set of branch
points, which conjugates dynamics of Lozi attractors to shifts on
inverse limits of such trees. A related result holds for Hénon
family within the intersection of Benedicks-Carleson and
Crovisier-Pujals parameter sets.
References:

- S. Crovisier, E. Pujals,
*Strongly dissipative surface
diffeomorphisms*, Commentarii Mathematici Helvetici 93 (2018),
377–400.
- S. Crovisier, E. Pujals, C. Tresser,
*Mild dissipative
diffeomorphisms of the disk with zero entropy*, arXiv 2020.
- R. Lozi,
*Un attracteur etrange(?) du type attracteur de Hénon*,
J. Physique (Paris) 39 (Coll. C5) (1978), 9–10.
- M. Misiurewicz,
*Strange attractor for the Lozi mappings*, Ann. New
York Acad. Sci. 357 (1980), 348–358.
- Topological and Smooth Dynamics on Surfaces, Mathematisches
Forschungsinstitut Oberwolfach Report No. 27/2020, DOI:
10.4171/OWR/2020/27