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In this article we study the Ahlfors regular conformal gauge of a compact metric space , and its conformal dimension . Using a sequence of finite coverings of , we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute using the critical exponent associated to the combinatorial modulus.
A planar polygonal billiard is said to have the finite blocking property if
for every pair of points in there exists a finite number of
“blocking” points such that every billiard trajectory from to
meets one of the ’s. Generalizing our construction of a counter-example to a
theorem of Hiemer and Snurnikov, we show that the only regular polygons that have the
finite blocking property are the square, the equilateral triangle and the hexagon. Then
we extend this result to translation surfaces....
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