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:

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