Pair correlations of aperiodic inflation rules via renormalisation: Some interesting examples

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• 作者：Michael Baake ; ^{mbaake@math.uni-bielefeld.de" class="auth_mail" title="E-mail the corresponding author} ; Franz Gä ; hler ^{gaehler@math.uni-bielefeld.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：37B10 ; 42A38 ; 52C23
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 June 2016
• 年：2016
• 卷：205
• 期：Complete
• 页码：4-27
• 全文大小：595 K

文摘

This article presents, in an illustrative fashion, a first step towards an extension of the spectral theory of constant length substitutions. Our starting point is the general observation that the symbolic picture (as defined by the substitution rule) and its geometric counterpart with natural prototile sizes (as defined by the induced inflation rule) may differ considerably. On the geometric side, an aperiodic inflation system possesses a set of exact renormalisation relations for its pair correlation coefficients. Here, we derive these relations for some paradigmatic examples and infer various spectral consequences. In particular, we consider the Fibonacci chain, revisit the Thue–Morse and the Rudin–Shapiro system, and finally analyse a twisted extension of the silver mean chain with mixed singular spectrum.