 
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 onedimensional 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 onedimensional ergodic
Schr¨odinger operators. Those are bounded selfadjoint operators of 2
(Z) given by
(1.1) (Hu)n = un+1 + un1 + 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 µalmost every x. In this case, the spectrum
is µalmost surely independent of x. The Lyapunov exponent is defined as
