Ergodic Theory and Applications in Combinatorial Number Theory

摘要：本文主要研究区间交换变换（iet’s）和区间交换映射（itm’s, iet’s的非双射版本）的丢色图、组合和谱性质。与其他人的“全局”方法不同，Boshernitan的方法特别强调所涉及的参数的丢芬图和代数性质可能影响一个对象（或一个项目）的动态性质的方式。（考虑的丢番图属性通常是“通用的”，导致“全局的”结果。）对于一些拓扑熵为零的动力系统，需要对相应的“块增长”进行精确的渐近估计，这种估计优于次指数估计，在某些情况下甚至是多项式估计。这些问题通常是困难的，比测量理论的设置更困难，并且目前在这个方向上的许多可用的积极结果（Katok, Yomdin, Gutkin, Haydn， Goetz和其他人）通常被认为远非最好的可能。例如，即使对于items，关于线性块增长的猜想（这是由计算机计算支持的）仍然是开放的，并且猜想的肯定解决将具有重要的后果。Boshernitzan还参与了低（特别是线性）复杂性的动力系统（符号流）的研究，以及在Ferenczi， casassaigne， Durand等人的工作中发现的针对这种背景的独特遍历性概念的各种强化。尽管这个项目的重点是在低复杂性的动力系统研究中出现的理论问题，但它与其他数学领域（数论、概率论和组合学，仅举三例），以及统计学、计算机科学（复杂性理论）和物理学（量子力学）有着许多自然的联系。Boshernitzan和他的合作者Kornfeld最近发现的“奇异的”区间平移图可能被证明在激光技术中有应用。

Proposal: DMS-9971120Principal Investigator: Michael D. BoshernitzanAbstract: The central subject of this proposal is diophantine, combinatorial and spectral properties of iet's (interval exchange transformations) and itm's (interval exchange maps, the nonbijective version of iet's). Boshernitan's approach, unlike the "global" approaches of others, puts special emphasis on the way diophantine and algebraic properties of the parameters involved may affect the dynamical properties of an iet (or an itm). (The diophantine properties considered are often "generic," leading to "global" results.) For some dynamical systems of topological entropy zero, an accurate asymptotic estimate on the corresponding "block growth" is needed that is better than merely subexponential, and in some cases, is even polynomial. These problems are often difficult, more difficult than in the measure-theoretic setting, and many presently available positive results in this direction (of Katok, Yomdin, Gutkin, Haydn, Goetz, and others) are generally believed to be far from best possible. For example, even for itm's, the conjecture on linear block growth (which is supported by computer computations) remains open, and an affirmative resolution of the conjecture would have important consequences. Boshernitzan is also involved in the study of dynamical systems (symbolic flows) of low (in particular, of linear) complexity, as well as various strengthenings of the notion of unique ergodicity tailored to this context, as found in the work of Ferenczi, Cassaigne, Durand, and others. Even though the emphasis of this project is on theoretical questions that arise in the study of dynamical systems of low complexity, there are numerous natural connections with other fields of mathematics (number theory, probability, and combinatorics, to name three), as well as with statistics, computer science (complexity theory, for one) and physics (quantum mechanics). The recent discovery made by Boshernitzan and his collaborator Kornfeld of "exotic" interval translation maps may prove to have applications in laser technology.