Ergodic Theory of d-adic Endomorphisms

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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.