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Continuity of Lyapunov Exponents via Large Deviations

Presented by: 
S Klein [Department of Mathematical Sciences, NTNU]
Date: 
Thursday 2nd April 2015 - 12:30 to 13:30
Venue: 
INI Seminar Room 2
Abstract: 
Large deviation type (LDT) estimates for transfer matrices are important tools in the study of discrete, one dimensional, quasi-periodic Schrodinger operators. They have been used to establish positivity of the Lyapunov exponent, continuity properties of the Lyapunov exponent and of the integrated density of states, estimates on the Green's function, Anderson localization. We prove - in a general, abstract setting - that the availability of appropriate LDT estimates implies continuity of the Lyapunov exponents, with a modulus of continuity depending explicitly on the strength of the LDT. The devil is of course in the details, hidden here behind the words "availability" and "appropriate". We show that the study of the Lyapunov exponents associated with a band lattice quasi-periodic Schrodinger operator fits this abstract setting, provided the potential is a real analytic function of (one or of) several variables and that the frequency vector is Diophantine. Co-authored with: P Duarte

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons