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RECENT DEVELOPMENTS IN MATHEMATICAL QUANTUM CHAOS
, 2009
"... This is a survey of recent results on quantum ergodicity, specifically on the large energy limits of matrix elements relative to eigenfunctions of the Laplacian. It is mainly devoted to QUE (quantum unique ergodicity) results, i.e. results on the possible existence of a sparse subsequence of eigen ..."
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This is a survey of recent results on quantum ergodicity, specifically on the large energy limits of matrix elements relative to eigenfunctions of the Laplacian. It is mainly devoted to QUE (quantum unique ergodicity) results, i.e. results on the possible existence of a sparse subsequence of eigenfunctions with anomalous concentration. We cover the lower bounds on entropies of quantum limit measures due to Anantharaman, Nonnenmacher, and Rivière on compact Riemannian manifolds with Anosov flow. These lower bounds give new constraints on the possible quantum limits. We also cover the nonQUE result of Hassell in the case of the Bunimovich stadium. We include some discussion of Hecke eigenfunctions and recent results of Soundararajan completing Lindenstrauss ’ QUE result, in the context of matrix elements for Fourier integral operators. Finally, in answer to the potential question ‘why study matrix elements’ it presents an application of the author to the geometry of nodal sets.
On the AubryMather theory for symbolic dynamics
, 2008
"... We propose a new model of ergodic optimization for expanding dynamical systems: the holonomic setting. In fact, we introduce an extension of the standard model used in this theory. The formulation we consider here is quite natural if one wants a meaning for possible variations of a real trajectory u ..."
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We propose a new model of ergodic optimization for expanding dynamical systems: the holonomic setting. In fact, we introduce an extension of the standard model used in this theory. The formulation we consider here is quite natural if one wants a meaning for possible variations of a real trajectory under the forward shift. In another contexts (for twist maps, for instance), this property appears in a crucial way. A version of the AubryMather theory for symbolic dynamics is introduced. We are mainly interested here in problems related to the properties of maximizing probabilities for the twosided shift. Under the transitive hypothesis, we show the existence of subactions for Hölder potentials also in the holonomic setting. We analyze then connections between calibrated subactions and the Mañé potential. A representation formula for calibrated subactions is presented, which drives us naturally to a classification theorem for these subactions. We also investigate properties of the support of maximizing probabilities. 1
Ergodic Optimization
, 2011
"... The field is a relatively recently established subfield of ergodic theory, and has significant input from the two wellestablished areas of symbolic dynamics and Lagrangiandynamics.Thelargescalepictureofthefieldis that one is interested in optimizing potential functions over the (typically highly c ..."
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The field is a relatively recently established subfield of ergodic theory, and has significant input from the two wellestablished areas of symbolic dynamics and Lagrangiandynamics.Thelargescalepictureofthefieldis that one is interested in optimizing potential functions over the (typically highly complex) class of invariant measures for a dynamical system. Tools that have been employed in this area come from convex analysis, statistical physics, probability theory and dynamic programming. The field also has both general aspects (in which the optimization is considered in the large on whole Banachspaces)andlocalaspects(inwhich the optimization is studied on individual functions). In the lattercategory,therehasbeeninputfromphysicists with numerical simulations suggesting that the optimizing measures are typically supported on periodic orbits. This should be contrasted with the situation typically found in the ‘thermodynamic formalism ’ of ergodic theory, in which the measures picked out by variational principles tend to have wide support. Ergodic optimization may be viewed as the low temperature limit of thermodynamic formalism. 2 Recent Developments and Open Problems Recent developments in the field have been multifaceted: there has been a general goal of establishing results showing that for a typical potential function, the optimizing measures are supported on periodic orbits. Until now, all results of this type have been established on separable Banach spaces of functions, whereas the principal
MEASURABLE DYNAMICS: THEORY AND APPLICATIONS Christopher Bose (University of Victoria),
, 2006
"... 1 Brief overview of the field The central aim of measurable dynamics is to apply modern mathematical techniques, including measure and probability theory, topology and functional analysis to study the timeevolution of complex evolving systems. The fact that many simple models in the natural science ..."
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1 Brief overview of the field The central aim of measurable dynamics is to apply modern mathematical techniques, including measure and probability theory, topology and functional analysis to study the timeevolution of complex evolving systems. The fact that many simple models in the natural sciences may lead to classically intractable mathematical problems was already observed in the 19th century by H. Poincare ́ during his investigations into the orbits of celestial bodies. At about the same time, the formal development of thermodynamic theory alerted scientists to a major shift in the mathematical modelling paradigm that was about to take place. Since then, researchers have coined terms like chaos and strange attractor to describe the perplexing properties observed by Poincaré and others, and we now know that these systems, rather than being isolated curiosities are, in fact, increasingly likely to be encountered once one leaves the familiar territory of standard mathematical models derived from classical Physics, Chemistry or Engineering. While the origins of the field are rooted in application, the mathematical development in the next century embraced both theoretical and applied approaches. In fact, for the first half of the 20th century, it is fair to say the former dominated as mathematicians struggled to develop new tools to describe the complex systems they were encountering. The celebrated ergodic theorems of Birkhoff and von Neumann, the development of