Quantum Unique Ergodicity and Rigidity in Dynamical Systems

New York University, New York NY

Investigators

Abstract

PI: Lindenstrauss Proposal Number: 0140497 ABSTRACT Quantum Unique Ergodicity and Rigidity in Dynamical Systems The research proposed lies at the interface of dynamical systems and several other mathematical disciplines, and in particular number theory and the mathematical theory of Quantum Chaos. It is well known that the collections of invariant probability measures and closed invariant sets for hyperbolic maps or flows is very large; remarkably, in many dynamical systems of algebraic origin where there are two (or more) commuting hyperbolic maps or flows it is conjectured that there are actually very few measures or closed sets invariant under this bigger action. Despite (or perhaps because of) important contributions by several authors, the mystery of this rigidity property of multidimensional actions is still one of the central problems in modern ergodic theory.While seemingly unrelated, the methodology of the study of rigidity of multiparameter actions is very suitable to study the arithmetical case of the Quantum Unique Ergodicity Conjecture, regarding the limit of the spatial distribution of a free particle on certain manifolds in a quantum steady state as the energy of the state tends to infinity (the semiclassical limit). This question has been considered previously by many authors mostly using tools from analytic number theory; the approach proposed, using the dynamical approach, is new and has already borne fruits. The theory of dynamical systems gives tools to study complex systems, for example the evolution of a complicated deterministic process over time. There is a rich and distinguished tradition of using these powerful tools in other mathematical disciplines, most notably in number theory and combinatorics. The research proposed is connected to several of the most exciting possible applications, including new implications to the theory of quantum chaos in mathematical physics.