 
Summary: Schršodinger operators with RudinShapiro potentials are not
palindromic
J.P. Allouche
CNRS, LRI, Batiment 490, F91405 Orsay Cedex (France)
allouche@lri.lri.fr
Abstract We prove a conjecture of A. Hof, O. Knill and B. Simon by showing that the Rudin
Shapiro sequence is not palindromic, i.e., does not contain arbitrarily long palindromes. We prove
actually this property for all paperfolding sequences and all RudinShapiro sequences deduced from
paperfolding sequences. As a consequence and as guessed by the above authors, their method
cannot be used for establishing that discrete Schršodinger operators with RudinShapiro potentials
have a purely singular continuous spectrum.
I. Introduction
In [1] the authors study the spectrum of discrete Schršodinger operators with potentials taking
values in a finite set of real numbers. They prove inter alia the following: suppose that the sequence
of potentials generates a subshift X that is strictly ergodic (i.e., both minimal and uniquely ergodic).
Then, a sufficient condition ensuring the existence of a generic set in X for which the operator has
a purely singular continuous spectrum, is that there exists a z in X that is palindromic, i.e. that
contains arbitrarily long palindromes.
This result covers many cases where X is generated by the fixed point of a primitive substitution
(Fibonacci, ProuhetThueMorse, perioddoubling, binary and ternary nonPisot) and also the case
