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Summary: ON THE SPECTRUM AND LYAPUNOV EXPONENT OF LIMIT
PERIODIC SCHR¨ODINGER OPERATORS
ARTUR AVILA
Abstract. We exhibit a dense set of limit periodic potentials for which the
corresponding one-dimensional Schr¨odinger operator has a positive Lyapunov
exponent for all energies and a spectrum of zero Lebesgue measure. No ex-
ample with those properties was previously known, even in the larger class of
ergodic potentials. We also conclude that the generic limit periodic potential
has a spectrum of zero Lebesgue measure.
1. Introduction
This work is motivated by a question in the theory of one-dimensional ergodic
Schr¨odinger operators. Those are bounded self-adjoint operators of 2
(Z) given by
(1.1) (Hu)n = un+1 + un-1 + v(fn
(x))un,
where f : X X is an invertible measurable transformation preserving an ergodic
probability measure µ and v : X R is a bounded measurable function, called the
potential.
One is interested in the behavior for µ-almost every x. In this case, the spectrum
is µ-almost surely independent of x. The Lyapunov exponent is defined as
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