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Đề tài " Hausdorff dimension of the set of nonergodic directions "

It is known that nonergodic directions in a rational billiard form a subset of the unit circle with Hausdorff dimension at most 1/2. Explicit examples realizing the dimension 1/2 are constructed using Diophantine numbers and continued fractions. A lower estimate on the number of primitive lattice points in certain subsets of the plane is used in the construction. | Annals of Mathematics Hausdorff dimension of the set of nonergodic directions By Yitwah Cheung Annals of Mathematics 158 2003 661 678 Hausdorff dimension of the set of nonergodic directions By Yitwah Cheung with an Appendix by M. Boshernitzan Abstract It is known that nonergodic directions in a rational billiard form a subset of the unit circle with Hausdorff dimension at most 1 2. Explicit examples realizing the dimension 1 2 are constructed using Diophantine numbers and continued fractions. A lower estimate on the number of primitive lattice points in certain subsets of the plane is used in the construction. 1. Introduction Consider the billiard in a polygon Q. A fundamental result KMS implies that a typical trajectory with typical initial direction will be equidistributed provided the angles of Q are rational multiples of n. More precisely there is a flat surface X associated to the polygon such that each direction 0 G S1 determines an area-preserving flow on X the assertion is that the set NE Q of parameters 0 for which the associated flow is not ergodic has measure zero. The statement holds more generally for the class of rational billiards in which the abstract polygon is assumed to have the property that the subgroup of O 2 generated by the linear parts of the reflections in the sides is finite. For a recent survey of rational billiards see MT . Let Qx A G 0 1 be the polygon described informally as a 2-by-1 rectangle with an interior wall extending orthogonally from the midpoint of a longer side so that its distance from the opposite side is exactly A see Figure 1 . We are interested in the Hausdorff dimension of the set NE Qa . Recall that A is Diophantine if the inequality A - - q q e has at most finitely many integer solutions for some exponent e 0. 662 YITWAH CHEUNG Figure 1. The billiard in Qx. Theorem 1. If X is Diophantine then H.dimNE Qx 1 2. In fact Masur has shown that for any rational billiard the set of nonergodic directions has Hausdorff .