Uniform ergodic theorems for discontinuous skew-product flows and applications to Schrödinger equations

@article{Zhang2011UniformET,
title={Uniform ergodic theorems for discontinuous skew-product flows and applications to Schr{\"o}dinger equations},
author={Meirong Zhang and Zhe Zhou},
journal={Nonlinearity},
year={2011},
volume={24},
pages={1539 - 1564}
}
• Published 1 May 2011
• Mathematics
• Nonlinearity
Motivated by linear Schrödinger equations with almost periodic potentials and phase transitions over almost periodic lattices, we introduce the so-called skew-product quasi-flows (SPQFs), which may admit both temporal and spatial discontinuity. In this paper we establish two basic theorems for SPQFs. One is an extension of the Bogoliubov–Krylov theorem for the existence of invariant Borel probability measures and the other is the uniform ergodic theorems. As applications, it will be shown that…
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References

SHOWING 1-10 OF 38 REFERENCES
Rotation Numbers of Linear Schrödinger Equations with Almost Periodic Potentials and Phase Transmissions
• Mathematics
• 2010
In this paper we study the linear Schrödinger equation with an almost periodic potential and phase transmission. Based on the extended unique ergodic theorem by Johnson and Moser, we will show for
LYAPUNOV EXPONENTS AND SPECTRAL ANALYSIS OF ERGODIC SCHRÖDINGER OPERATORS: A SURVEY OF KOTANI THEORY AND ITS APPLICATIONS
The absolutely continuous spectrum of an ergodic family of onedimensional Schrödinger operators is completely determined by the Lyapunov exponent as shown by Ishii, Kotani and Pastur. Moreover, the
Lyapunov Exponents and Spectral Analysis of Ergodic Schrödinger Operators: A Survey of Kotani Theory and Its Applications
The absolutely continuous spectrum of an ergodic family of onedimensional Schrodinger operators is completely determined by the Lyapunov exponent as shown by Ishii, Kotani and Pastur. Moreover, the
Semi-uniform ergodic theorems and applications to forced systems
• Mathematics
• 2000
In nonlinear dynamics an important distinction exists between uniform bounds on growth rates, as in the definition of hyperbolic sets, and non-uniform bounds as in the theory of Liapunov exponents.
Reducibility or nonuniform hyperbolicity for quasiperiodic Schrodinger cocycles
• Mathematics
• 2003
We show that for almost every frequency ?? ?? R\Q, for every C?O potential v : R/Z ?? R, and for almost every energy E the corresponding quasiperiodic Schr?Nodinger cocycle is either reducible or
Almost Periodic Differential Equations
Almost periodic functions.- Uniformly almost periodic families.- The fourier series theory.- Modules and exponents.- Linear constant coefficient equations.- Linear almost periodic equations.-
An ergodic theorem for Delone dynamical systems and existence of the integrated density of states
• Mathematics
• 2003
We study strictly ergodic Delone dynamical systems and prove an ergodic theorem for Banach space valued functions on the associated set of pattern classes. As an application, we prove existence of
The rigidity of reducibility of cocycles on
• Mathematics
• 2008
In this paper, we prove that for almost all and constant , if an analytic one-dimensional quasi-periodic cocycle (α, A) on is conjugated to the constant cocycle (α, C) by a measurable conjugacy (0,
Introduction to Ergodic Theory
Ergodic theory concerns with the study of the long-time behavior of a dynamical system. An interesting result known as Birkhoff’s ergodic theorem states that under certain conditions, the time
Exponential growth of product of matrices in
• Mathematics
• 2008
In this paper we investigate the exponential growth of products of two matrices . We prove, assuming A is a fixed hyperbolic matrix, that for Lebesgue almost every B, products of length n involving