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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 f_{a,b}(x,y)=(1-a|x|+by,x) on \mathbb R^2, small numbers \epsilon_1,\epsilon_2>0 such that, setting (a,b)=(1.4+\epsilon_1,0.4+\epsilon_1), for all 0<\epsilon<\epsilon_2:

  • h_{top}(f_{a,b})=0;
  • h_{top}(f_{a+\epsilon,b})>\frac14\log\frac12(\sqrt{5}+1).

The verification turns out to be quite simple (once you know where to look!). The non-wandering set of f_{a,b} is shown to be reduced to be reduced to the fixed points of its fourth iterates, yielding the zero entropy immediately. f_{a+\epsilon,b} on the other hand is shown to admit 2 disjoint closed quadrilaterals U,V such that f^4(U) hyperbolically crosses both U and V and f^4(V) hyperbolically crosses U. This means that the sides of U and V can be branded alternatively s and u with the following property. The image of a u side crosses each of U,V it meets, intersecting both their s sides and none of their u sides. This again yields the entropy estimate.

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