Abstract view
Pure Discrete Spectrum for Onedimensional Substitution Systems of Pisot Type


Published:20021201
Printed: Dec 2002
V. F. Sirvent
B. Solomyak
Abstract
We consider two dynamical systems associated with a substitution of
Pisot type: the usual $\mathbb{Z}$action on a sequence space, and
the $\mathbb{R}$action, which can be defined as a tiling dynamical
system or as a suspension flow. We describe procedures for checking
when these systems have pure discrete spectrum (the ``balanced
pairs algorithm'' and the ``overlap algorithm'') and study the
relation between them. In particular, we show that pure discrete
spectrum for the $\mathbb{R}$action implies pure discrete spectrum
for the $\mathbb{Z}$action, and obtain a partial result in the
other direction. As a corollary, we prove pure discrete spectrum
for every $\mathbb{R}$action associated with a twosymbol
substitution of Pisot type (this is conjectured for an arbitrary
number of symbols).
MSC Classifications: 
37A30, 52C23, 37B10 show english descriptions
Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} Quasicrystals, aperiodic tilings Symbolic dynamics [See also 37Cxx, 37Dxx]
37A30  Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} 52C23  Quasicrystals, aperiodic tilings 37B10  Symbolic dynamics [See also 37Cxx, 37Dxx]
