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Schrodinger operators with RudinShapiro potentials are not palindromic

Summary: Schršodinger operators with Rudin­Shapiro potentials are not
J.­P. Allouche
CNRS, LRI, B“atiment 490, F­91405 Orsay Cedex (France)
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


Source: Allouche, Jean-Paul - Laboratoire de Recherche en Informatique, Université de Paris-Sud 11


Collections: Computer Technologies and Information Sciences