Problems in Measurable Dynamics

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A range of studies in measurable dynamics of are proposed.They lie generally in extensions and applications of orbit equivalence methods. The PIhas helped to develop two general approaches here. The first is Restricted Orbit Equivalence, whereone places restrictions on the sorts of orbit preserving maps allowed. The second,and new, method is an "orbit transference method" where the orbit equivalence is arbitrarybut measurable with respect to an algebra orthogonal to the algebra of interest. Among theapplications proposed for these methods are:1) a deeper and perhaps complete understanding the the Weak Pinsker property of Thouvenot2) a version of the theory of Isometric Extensions of Bernoulli automorphism for some class of restricted orbit equivalences, in particular for standard dyadic reverse filtrations.3) extensions of restricted orbit equivalence beyond actions of descrete amenable groups to spaces of labeled random graphs, to not necessarilly descrete groups and to non-singular actions. Most important here is that one should look for good examples, for example an entropy theory for nonsingular actions.Large scale phenomena can often exhibit a random behavior. Crystals often include random inclusions. Large data bases often (always?) include random errors. The genome can be modeled as an infinite sequence of base pairs that in places is random and places not so. One models such structures in probabilistic terms as a space of linked nodes to which are attached labels, colors if you will, endowed with some notion of the probability of occurance of some particular local labeling or picture occuring within an infinite picture. It is natural to look for ways of coding and comparing such random arrays among themselves. One can seek extremely rigid encodings that allow no distortion of the nodes and their links, just an encoding of the label at a particular node to that same node in the other labelingof the array. Or more generally and interestingly one can allow the nodes and links to distort as well. By setting some control on the distortions one can craft an approach to understanding a variety of properties of such systems. Although the goals of this proposal are of an abstract nature, one of the first such controlled distortions (called f-bar) was brought over from genetics where it was used as a measure of evolutionary relatedness among DNA molecules. What is proposed here are a variety of applications of methods involving controlled distortion of such labeled arrays to understand such random phenomena more deeply.