Meirong Zhang and Zhe Zhou 2011 Nonlinearity 24 1539 doi:10.1088/0951-7715/24/5/008
Meirong Zhang1,2 and Zhe Zhou1,3
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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 such a Schrödinger equation admits a well-defined rotation number.
05.20.-y Classical statistical mechanics
03.65.Ge Solutions of wave equations: bound states
37A20 Orbit equivalence, cocycles, ergodic equivalence relations
37E45 Rotation numbers and vectors
47E05 Ordinary differential operators (See also 34Bxx, 34Lxx)
Issue 5 (May 2011)
Received 2 September 2010, in final form 7 March 2011
Published 1 April 2011
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