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On the conformal gauge of a compact metric space

Matias Carrasco Piaggio (2013)

Annales scientifiques de l'École Normale Supérieure

In this article we study the Ahlfors regular conformal gauge of a compact metric space ( X , d ) , and its conformal dimension dim A R ( X , d ) . Using a sequence of finite coverings of  ( X , d ) , 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 dim A R ( X , d ) using the critical exponent Q N associated to the combinatorial modulus.

On the finite blocking property

Thierry Monteil (2005)

Annales de l’institut Fourier

A planar polygonal billiard 𝒫 is said to have the finite blocking property if for every pair ( O , A ) of points in 𝒫 there exists a finite number of “blocking” points B 1 , , B n such that every billiard trajectory from O to A meets one of the B i ’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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