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Pure Discrete Spectrum for One-dimensional Substitution Systems of Pisot Type

  Published:2002-12-01
 Printed: Dec 2002
  • V. F. Sirvent
  • B. Solomyak
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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 two-symbol 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]
 

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