Ergodic Theory of d-adic Endomorphisms

d-进自同态的遍历理论

• 批准号: 0100445
• 负责人: Christopher Hoffman
• 金额: $ 9.14万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100445&HistoricalAwards=false
• 关键词: Ergodic; Theory; adic; Endomorphisms

Hoffman will investigate the properties of endomorphisms which areisomorphic to one sided Bernoulli shifts. One direction this workwill take is to show that particular dynamical systems are isomorphicto one sided Bernoulli shifts. In particular Hoffman will attempt toclassify which toral endomorphisms are isomorphic to a one sidedBernoulli shift. He will also attempt to classify all toralendomorphisms up to isomorphism. Hoffman will try to show thedifferences between Ornstein's theory of Bernoulli automorphisms andthe emerging theory of Bernoulli endomorphisms. A particular exampleof this relates to compact extensions of Bernoulli endomorphisms.Rudolph proved that any compact extension of a Bernoulli shift that isweak mixing is isomorphic to a Bernoulli shift. Hoffman will try toshow that Bernoulli endomorphisms do not share this property. Hoffmanwill also continue his work in probability theory. He will continueto study infinite percolation cluster on the square lattice. Inparticular he will compare the properties of simple random walk on thesquare lattice with the properties of simple random walk on infinitepercolation clusters. The overall goal of this portion of theresearch is to determine if the distribution of paths for simplerandom walk on the infinite percolation cluster can be rescaled insuch a way as to generate the distribution of Brownian paths in theplane.The study of randomness is a major part of the application ofmathematics to many interesting systems. For example, in statisticsknowledge of randomness is used to make sense of data and in appliedmath random processes are used to model financial markets dynamicalsystems. This proposal deals with randomness in the context ofdynamical systems and probability theory. This proposal will attemptto classify various types of mathematical systems according to thetype and amount of randomness that they exhibit.

霍夫曼将研究同构于单侧伯努利移位的自同态的性质。 这项工作的一个方向是表明，特定的动力系统是同构的单边伯努利位移。特别是霍夫曼将试图分类哪些toral自同态同构的一个sidedBernoulli移位。 他还将尝试分类所有toralendomorphisms到同构。 霍夫曼将试图显示奥恩斯坦的伯努利自同构理论和新兴的伯努利自同态理论之间的差异。 一个特别的例子涉及到Bernoulli自同态的紧扩张。Rudolph证明了任何弱混合的Bernoulli移位的紧扩张都同构于Bernoulli移位。 霍夫曼将试图表明，伯努利自同态不共享此属性。 霍夫曼也将继续他的工作在概率论。 他将继续研究正方形格子上的无限渗流集团。 特别是，他将比较性质的简单随机行走的正方形格子上的性质的简单随机行走的无限渗流集群。 这部分研究的总体目标是确定无限渗流簇上简单随机行走的路径分布是否可以重新缩放，从而生成平面上的布朗路径分布。随机性研究是数学应用于许多有趣系统的主要部分。 例如，在数学中，随机性的知识被用来理解数据，在应用数学中，随机过程被用来模拟金融市场的动态系统。 这一建议涉及随机性的背景下，动力系统和概率论。 这一建议将根据数学系统所表现出的随机性的类型和数量来对各种类型的数学系统进行分类。