Hoffman will investigate the properties of endomorphisms which are isomorphic to one sided Bernoulli shifts. One direction this work will take is to show that particular dynamical systems are isomorphic to one sided Bernoulli shifts. In particular Hoffman will attempt to classify which toral endomorphisms are isomorphic to a one sided Bernoulli shift. He will also attempt to classify all toral endomorphisms up to isomorphism. Hoffman will try to show the differences between Ornstein's theory of Bernoulli automorphisms and the emerging theory of Bernoulli endomorphisms. A particular example of this relates to compact extensions of Bernoulli endomorphisms. Rudolph proved that any compact extension of a Bernoulli shift that is weak mixing is isomorphic to a Bernoulli shift. Hoffman will try to show that Bernoulli endomorphisms do not share this property. Hoffman will also continue his work in probability theory. He will continue to study infinite percolation cluster on the square lattice. In particular he will compare the properties of simple random walk on the square lattice with the properties of simple random walk on infinite percolation clusters. The overall goal of this portion of the research is to determine if the distribution of paths for simple random walk on the infinite percolation cluster can be rescaled in such a way as to generate the distribution of Brownian paths in the plane.
The study of randomness is a major part of the application of mathematics to many interesting systems. For example, in statistics knowledge of randomness is used to make sense of data and in applied math random processes are used to model financial markets dynamical systems. This proposal deals with randomness in the context of dynamical systems and probability theory. This proposal will attempt to classify various types of mathematical systems according to the type and amount of randomness that they exhibit.
