TAILIEUCHUNG - 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 Qx 1 2. In fact Masur has shown that for any rational billiard the set of nonergodic directions has Hausdorff .

• Thạc sĩ - Tiến sĩ - Cao học
• Hausdorff dimension
• subjects of mathematics
• Annals of Mathematics
• science reports
• studied mathematics
• natural sciences
• scientific research
• Research methods
• Data collection methods
• Understanding scientific research
• Ensuring research validity
• Measurement techniques
• Sampling methods
• Scientific research capacity
• Educational institutions
• Policy and management studies
• Vietnam Academy for Ethnic minorities
• Scientific bases
• earth sciences
• Research
• VNU Journal of Science
• scientific reports
• Action research
• linguistics
• research projects
• Foreign Languages
• cooperative research
• Renewable energy
• Introduction to research
• Scientific investigation
• Scientific investigation research
• Experimental designs
• Measurement of variables
• Reading research
• Conducting and reading research
• Research process
• Research report
• Purpose of research
• Scientific method
• Conducting research
• The effect of university autonomy on training and scientific research
• Vietnam national university of agriculture
• Training and scientific research
• Vascular plant
• Flora
• Yen Tu
• Quang Ninh Research Forest
• statements and documents scientific reports
• medical reports
• medical knowledge
• medical research
• clinical trials
• medical management
• health care
• Methods in behavioral research
• Behavioral research
• Scientific understanding
• Fundamental research issues
• Measurement concepts
• Observational methods
• reading comprehension
• education
• optimal picket
• model creation
• urban expansion
• organizing spatial
• Journal of Science
• Journal of Technology
• Technology Management
• Research quality
• Strategy and policy
• Summary of Doctoral thesis
• Summary of Doctoral thesis in public Management
• State management
• Female scientific research human
• Research human resource development
• Scientific research and teaching
• Defense and security education
• Current situation and solutions
• Security education
• Structural biochemistry
• epigenetics
• MicroRNAs
• Tạp chí khoa học
• Tạp chí Công nghệ
• Quản lý công nghệ
• Human resources
• Science management
• Poliomyelitis vaccine Research
• Non Ri Maprec
• Production center