Duncan SANDS gave a talk for the Journée Affine Par Morceaux on the dynamics of Lozi maps. These are the piecewise affine homeomorphisms of the plane of the form where . Lozi introduced them as a toy model for the Hénon map, observing numerically some kind of strange attractor for . SANDS and ISHII have especially studied their topological entropy. The following picture shows what is known and what is not in the parameter plane:
In the grey area the entropy is known to be zero. In the turquoise area it is known to be positive (and maximal, ie, log 2, in the hatched part). In the white area in-between, there are examples with positive entropy (on for the part below the axis) but otherwise little is known.
The existence of a physical measure has been established only for a small part of the Misiurewicz triangle (for which a strange attractor is known). There is a larger triangle in which a simple trapping region exists.