I have shown that, like diffeomorphisms, piecewise affine surface homeomorphisms are approximated in entropy by horseshoes, away from their singularities. It follows in particular that their topological entropy is lower-semicontinuous: a small perturbation cannot cause a macroscopic drop in entropy.
The continuity of the entropy for such maps had been an open problem for some time. Rigorous numerical estimates by Duncan SANDS and Yutaka ISHII seemed to suggest some discontinuous drops, but investigation at a small scale suggested these drops to be steep yet continuous variations.
Izzet B. YILDIZ has solved this question by finding for Lozi maps 
;
.
The verification turns out to be quite simple (once you know where to look!). The non-wandering set of 
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:
for the part below the axis) but otherwise little is known.
Jérôme Buzzi's 