Ergodic Theory

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Abstract Rudolph Over the coming three years Rudolph proposes to focus efforts in three directions. Having proved the multi-term return-times theorem by way of multi-fold genericity of sequences for measures, he now proposes to consider multiple recurrence results. Having just completed with Janet Kammeyer a multi-year project to lay the foundations of restricted orbit equivalence for discrete amenable groups, he proposes to apply this machinery to classical and novel problems. In particular, the application of restricted orbit equivalence methods to Vershik's theory of reverse filtrations has been extremely fruitful and will be continued. Finally, he proposes efforts on constructive problems, in particular exotic Markov randomfields and smooth examples related to the classical T,T-inverse map. Most if not all interesting phenomena have some random character. Ergodic theory, as envisioned in this project, is the attempt to ferret out the nature and origin of this randomness in phenomena modeled abstractly as probabilistic processes. This is done by examining the behavior of the process over ever larger blocks of time and attempting to analyze or recode the information gained. The principle investigator has developed an approach to this analysis that considers controlled distortions of time in the process to attempt to modify it to some simpler form. The degree or manner of distortion one allows will determine what properties of the process are preserved or lost. This allows one to focus on some particular properties to save allowing others to be eliminated. One particularly interesting case of this is when randomness appears spatially in a crystal or tiling and "time" is two or three dimensional. The crystal structure or tile shapes will determine what "distortions" are allowed. This "time distortion" approach, called "restricted orbit equivalence" has already shown itself well-suited to understanding a wide variety of properties of random processes and appears applicable to many others. Pursuing these possibilities is the main focus of this project.