Ergodic Theory of d-adic Endomorphisms

Hoffman将研究同构于单边Bernoulli移位的自同态的性质。这项工作将采取的一个方向是证明特定的动力系统同构于单边伯努利位移。特别是，霍夫曼将尝试对哪些同构于单边伯努利移位的自同态进行分类。他还将尝试将所有的自同构归类到同构。霍夫曼将试图说明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区别。一个特例涉及Bernoulli自同态的紧扩张。Rudolph证明了弱混合的Bernoulli位移的任何紧扩张都同构于Bernoulli位移。霍夫曼将试图证明伯努利自同态不共享这一性质。霍夫曼还将继续他在概率论方面的工作。他将继续研究正方形晶格上的无限渗流团簇。特别是，他将把方格上的简单随机游动的性质与无限渗流簇上的简单随机游动的性质进行比较。这部分研究的总体目标是确定无限渗流簇上简单随机游动的路径分布是否可以重新缩放，以产生平面上的布朗路径分布。随机性的研究是数学应用于许多有趣系统的主要部分。例如，在统计学中，随机性知识被用来解释数据，而在应用数学中，随机过程被用来对金融市场动态系统进行建模。这一建议涉及动力系统和概率论背景下的随机性。这项建议将试图根据各种数学系统表现出的随机性的类型和数量对它们进行分类。

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.