Ergodic Theory of d-adic Endomorphisms

霍夫曼将研究与单侧伯努利位移同构的自同态的性质。这项工作将采取的一个方向是表明特定的动力系统是同构的单侧伯努利位移。特别地，Hoffman将尝试分类哪些所有的自同构与单侧伯努利移位同构。他也将尝试把所有的托向自同构划分为同构。霍夫曼将试图展示奥恩斯坦的伯努利自同态理论和新兴的伯努利自同态理论之间的差异。这方面的一个特别的例子与伯努利自同态的紧拓有关。Rudolph证明了任何弱混合的伯努利位移的紧化扩展都与伯努利位移同构。霍夫曼将试图证明伯努利自同态不具有这种性质。霍夫曼还将继续他在概率论方面的工作。他将继续研究方阵上的无限渗透簇。特别地，他将比较方格上的简单随机漫步与无限渗透簇上的简单随机漫步的性质。这部分研究的总体目标是确定无限渗聚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。对随机性的研究是将数学应用于许多有趣的系统的一个主要部分。例如，在统计学中，随机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。本建议处理动力系统和概率论背景下的随机性。本提案将尝试根据它们所表现出的随机性的类型和数量对各种类型的数学系统进行分类。

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.