Mathematical Sciences: Ergodic Theory and Applications in Combinatorial Number Theory

本文的中心内容是区间交换变换的丢番图性质。与其他人(H.Masur，W.Veech等)的全局方法相反，Boshernitan在研究IET的方法中，特别强调了所涉及参数的丢番图和代数性质可能影响IET动力学性质的方式。解析数论中的许多结果都可以推广到IET的结果，例如Boshernitzan和C.Carroll的最新结果，即二次域上的极小IET必是伪Anosof(自相似)，可以看作是经典的Lagrange关于二次无理数连分式表示的周期性的结果的推广。也许在这个方向上最突出的猜想是试图将著名的Roth定理推广到IET，即关于有理数可以逼近的速率代数无理数的Roth定理。(对应的刘维尔定理的类比是正确的。)Boshernitzan在解决上述问题时可以使用的工具之一是他最近发现了动力系统中典型轨道的重现率与空间的Hausdorff维度之间的联系(量化版的庞加莱递归定理)。对于IET来说，这种联系是双向的。在许多其他情况下也是如此(Y.Peres，D.Ornstein，B.Weiss)。Boshernitzan的其他兴趣包括数论、多边形中的台球、关于整数子序列的遍历定理、Hardy field(与M.Wierdl、D.Berend、G.Kolesnik、I.Kornfeld和V.Berelson的联合项目)。由于本研究提案的基本性质，在这一点上很难准确评估可能应用的领域。在可能从我们的研究中受益的主题中，我只提到创建在计算机技术中有用的新代码压缩方案的可能性(与我们发现的量化版本的Poincare递归定理有关)，以及可能应用于激光技术(与I.Kornfeld关于区间平移映射的研究)。

Abstract Boshernitzan The central subject in this proposal is diophantine properties of iet's (interval exchange transformations). Boshernitan's approach in the study of iet's, contrary to the 'global' approaches of others (H.Masur, W.Veech etc.), makes special emphasis on the way diophantine and algebraic properties of the parameters involved may affect the dynamical properties of an iet. Many results in analytical number theory are extendable to iet's. For example, Boshernitzan's recent result, joint with C.Carroll, that minimal iet's over a quadratic field must be pseudo-Anosof (self-similar) can be viewed as an extension of the classical Lagrange's result on the periodicity of the continued fraction representations of quadratic irrationals. Perhaps the most outstanding conjecture in this direction is obtained from an attempt to extend to iet's the famous Roth's theorem on the rate algebraic irrationals can be approximated by the rationals. (The analogue of the corresponding Liouville's theorem is true.) One of the tools at Boshernitzan's disposal in tackling the above problems is his recent discovery of a connection between the rate of recurrence of typical orbits in dynamical systems and the Hausdorff dimension of the space (quantitative version of Poincare recurrence theorem). For iet's, this connection works both directions. The same is true in many other situations (Y.Peres, D.Ornstein, B.Weiss). The list of other Boshernitzan's interests includes number theory, billiards in polygons, ergodic theorems along subsequences of integers, Hardy fields (joint projects with M.Wierdl, D.Berend, G.Kolesnik, I.Kornfeld and V.Bergelson). Because of the basic nature of the present research proposal, it is hard at this point to assess accurately the fields of possible applications. Among the subject which may benefit from our research I would only mention the possibility of creation of new code compression schemes useful in computer technology ( related to our discovery of quantitative version of Poincare recurrence theorem) and possible applications to laser technology (research with I. Kornfeld on interval translation maps).