APPLICATIONS OF ANDERSON LOCALIZATION TO DYNAMICAL SYSTEMS AND EVOLUTIONARY PDE

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The proposal has two parts. The first part addresses positive entropy problem for smooth maps. The concept is as follows. Take a piece of material, say a ball--shaped piece of dough.  Transform its shape, say by flattering into a disk. Then gather it back roll and shape back into a ball. Repeat the process a number of times. Naturally, the components of the material, say salt and others, get evenly distributed per each cubic inch of the material. That is called "mixing". In physics if the rate of the mixing is high it can be measured by a quantity called entropy. The entropy equation is rather involved. An accurate evaluation of the entropy is a hard mathematical problem. A good news is that the entropy can be related to some geometrical characteristics of the transformation in question. For instance, it makes sense that the stronger one "stratches" the piece of the material in one direction, the "thinner" it gets in the complimentary direction. So, after gathering back the material, the "mixing" goes faster. Only for special transformations the entropy has been evaluated. For most natural examples the problem is wide open. In 1969 Soviet physicist B.Chirikov discovered a simple map of the square which serves as an important model in mechanics of particles, accelerators, plasma physics. The map was later named Chirikov map and also standard map. The map depends on its major parameter which measures its total intensity. Chirikov produced numerical simulations which show convincingly that if the intensity is not low, then the main portion of the square exhibits fast mixing, or as mathematicians put it "the entropy is positive". To evaluate the standard map entropy with mathematical certainty proved to be a very hard problem. In the current proposal we plan to develop a completely new method for standard map. The map naturally produces a system which belongs to completely different physics,-quantum mechanics. There is a conjecture that the entropy evaluation is hidden in certain special propertie of this quantum mechanical system.  In 1958 P.Anderson discovered absolutely new phenomenon in quantum mechanics. The phenomenon was later called Anderson localization. The discovery was one of the major citations for 1977 Nobel prize awarded to Anderson jointly with Mott and Van Vleck. We plan to use this phenomenon for entropy problem. The famous KdV equation was named after D.Korteweg and G. de Vries, who proposed it as a model for propagation of certain special waves. One needs to know the initial input to solve the equation. The solution method was developed for input periodic in space almost 40 years ago. Several years ago P.Deift , who is one of the leading experts in this field, stated a problem to develop a method for initial input which is not periodic but a mixture of a collection of periodic with different periods. In the second part of the proposal we plan to develop further the progress we made in this problem in last 8 years.