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Gibbs-like measure for spectrum of a class of quasi-crystals

Published online by Cambridge University Press:  14 January 2011

SHEN FAN ,
QING-HUI LIU  and
ZHI-YING WEN
SHEN FAN
Affiliation:
Department of Mathematics, Tsinghua University, Beijing, 100084, PR China (email: wenzy@mail.tsinghua.edu.cn)
QING-HUI LIU
Affiliation:
Department of Computer Science and Engineering, Beijing Institute of Technology, Beijing, PR China (email: qhliu@bit.edu.cn)
ZHI-YING WEN
Affiliation:
Department of Mathematics, Tsinghua University, Beijing, 100084, PR China (email: wenzy@mail.tsinghua.edu.cn)
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Abstract

Let α∈(0,1) be an irrational, and [0;a1,a2,…] the continued fraction expansion of α. Let Hα,V be the one-dimensional Schrödinger operator with Sturmian potential of frequency α. Suppose the potential strength V >20 and the sequence (ai)i≥1 is bounded. We proceed by developing some new ideas on dimensional theory of Cookie-cutter sets. We prove that the spectral generating bands satisfy the principles of bounded variation and bounded covariation, and then we show that there exists a Gibbs-like measure on the spectrum σ(Hα,V). As an application, we prove that where s* and s* are the lower and upper pre-dimensions. Moreover, if (an)n≥1 is ultimately periodic, then s* =s*.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 6 , December 2011 , pp. 1669 - 1695
Copyright
Copyright © Cambridge University Press 2011

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