Posts Tagged ‘surface dynamics’

Erwan Lanneau gave a talk in Orsay about his work on surfaces of translation. Pseudo-Anosov homeomorphisms are homeomorphisms of such surfaces which are affine away from the singular set of the surface and whose differential is hyperbolic. The dilation factor is the dominant eigenvalue of that differential. It is a Perron number.

The minimum of the dilation factor for given genus is known for geni 1 and 2 only (the techniques behind could be extended to genus 3 but not farther). One also knows that c_1/g\leq \log \delta(g)\leq c_2/g.

The main result of the talk is that the above does not hold when restricted to a given type of translation surfaces. More precisely, the moduli space of translation surfaces of given genus splits into connected components (at most three), one of them corresponding to hyperellipticity and the following holds:

Theorem (Boissy-Lanneau) Let \Phi be a pseudo-Anosov on a hyperelliptic translation surface of genus g admitting an involution with 2g+2 fixed points). Assume that \Phi has a unique singularity. Then its dilation is strictly greater than \sqrt{2} (but approach this value as g\to\infty).

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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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