Orbit Equivalences in Borel Dynamics

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Ergodic theory is the branch of mathematics that studies average behavior of dynamical systems, examples of which are movements of celestial bodies or gas particles. Exhaustive classification of all such systems is impossible, and it is fruitful to study notions of equivalence when dynamical systems share orbits but traverse them in a different order or share paths but follow them with different speeds. Such relations are called orbit equivalence relations. Due to their statistical nature, ergodic theoretical conclusions always leave the possibility of an exceptional behavior on some outlying orbits. Borel dynamics is a mathematical discipline that strives to adapt and enhance ergodic theoretical methods to study all orbits of a dynamical system, ruling out any exceptions. The PI will investigate variants of orbit equivalence within the scope of Borel dynamics. The project provides research training opportunities for graduate students.This project concentrates on the study of orbit equivalence relations of Borel flows. Complete classification of measure-preserving transformations is known to be an infeasible task, and ergodic theory has been advanced with the introduction of notions of equivalence that are weaker than the isomorphism. Notable examples include orbit equivalence, (even) Kakutani equivalence, and alpha equivalence, all of which can be viewed under the unified umbrella of restricted orbit equivalence formalism. The PI intends to investigate corresponding notions of orbit equivalence within the scope of Borel dynamics. Problems investigated in this project are likely to have connections with Borel combinatorics, ergodic theory, symbolic dynamics, and descriptive set theory of Polish group actions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.