Orbit Methods in Ergodic Theory

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The PI will study measurable orbit spaces. What is common to these spaces is that they are probability spaces; the points (or states) of such a space are linked into classes which one calls "orbits" and these orbits have a large-scale geometric or combinatorial structure. This structure could be the structure of the orbit of a countable or continuous group action or could be the structure of a binary tree of inverse images of some finite to one endomorphism. This structure could take the form of a topology or more strongly a metric on the orbits or could be a tree or graph structure with the states as nodes of the graph. The PI has developed a range of methods to investigate large-scale statistical properties on such spaces of orbits and to study notions of equivalence of these structures that allow for distortion or rearrangement of the orbits. The goal of this study is to extend these methods to as broad a perspective as possible and to apply them to answer interesting and significant questions. These methods have a significant history. They have allowed one to extend the Ornstein Isomorphism theory for Bernoulli automorphisms to endomorphisms (joint work with C. Hoffman) and give a tool for lifting large parts of the theory of actions of Z to actions of general discrete amenable groups (joint work with B. Weiss) via an orbit transference method. This proposal suggests an array of natural directions to proceed. The generalization to trees of inverse images of endomorphisms can be pushed to a study of trees or graphs in general and various geometric notions of similarity of such trees. Recent work indicates ways to effectively generalize to non-singular and even singular dynamics. One can also generalize both the labeling space and the index space to be continua, leading to a study of Brownian motion as the continuous analogue of a uniform endomorphism. Work to date, and the work proposed will continue to advance our understanding of measurable dynamics and of measurable orbit structures more generally. Such structures are common in mathematics and other hard sciences.Ergodic theory and measurable dynamics have their roots in thermodynamics, celestial mechanics, probability theory, and functional analysis. They have applications within mathematics to algebra, combinatorics, and number theory geometry, probability and statistics. Outside of mathematics, there are applications in physics, chemistry, electrical engineering and genetics. The training of graduate students is central to this proposal as well. The PI, the Department and the University of Maryland have demonstrated a strong commitment to diversity and to the proper training and mentoring of students.1