Unique Ergodic Theorem On Discontinuous Skew-product Flows And Rotation Numbers Of Linear Schr(?)dinger Equations

Models from classical mechanics are basic examples of dynamical systems. Oneof the main features is that these dynamical systems are continuous in temporal and spa-tial variables and are even di?erentiable. In this paper, motivated by linear Schro¨dingerequations with quantum e?ects, we will introduce the concept of skew-product quasi-?ows (SPQFs) which have both temporal and spatial discontinuity. To study the dy-namics of these equations, we will establish some ergodic results for SPQFs. Thispaper is divided into two parts: theory and application. Our main results are summa-rized in Chapter 1.Compared with classical skew-product ?ows, the discontinuity of SPQFs is anessential di?erence. In Chapter 2, in order to overcome the di?culties coming fromthe discontinuity, we firstly apply the fixed point theorem and the weak convergence ofmeasures to establish the existence theorem of invariant measures of SPQFs. Roughlyspeaking, when the set of discontinuous points is small, SPQFs still admit some in-variant Borel probability measures. Such a result is an extension of the celebratedBogoliubov-Krylov theorem. Applying further results from measure theory and topol-ogy, we will establish the unique ergodic theorem for SPQFs, which yields a uniformconvergence for the Birkho? temporal average. This is a complete extension of theunique ergodic theorem of Johnson and Moser.In Chapter 3, from point of view of compactness, we will give several equivalentdefinitions of almost periodic lattices, and analyze some dynamical properties corre-spondingly. This work plays a fundamental role in the following application.In Chapter 4, as an application of theory of SPQFs, we will establish the concept ofrotation numbers for linear Schro¨dinger equations with almost periodic potentials andphase transmissions on almost periodic lattices. One of the crucial steps is that, in orderto yield the corresponding SPQFs, besides the potential and the phase transmission, weadd the lattice into the phase space of SPQFs. Such an extension of phase spaces is a crucial breakthrough to establish rotation numbers. Meanwhile, we partially discussthe dependence of rotation numbers on the intensive parameter of quantum e?ects. Inour opinion, the present work has laid an important foundation for further study ofthese systems with quantum e?ects.