Gibbs-like measure for spectrum of a class of quasi-crystals
Gibbs-like measure for spectrum of a class of quasi-crystals
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DOI：
10.1017/s0143385710000635
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发表时间：
2011-01
影响因子：
0.9
通讯作者：
Shen Fan;Qinghui Liu;Z. Wen
中科院分区：
文献类型：
--
作者：
Shen Fan;Qinghui Liu;Z. Wen
摘要α∈（0,1）是一个非理性的，[0; a1，a2，…]α的持续分数为hα，v是具有频率α潜力的一维schrödinger潜在的强度v> 20和序列（AI）i≥1是有界的。协调，然后我们表明，在频谱σ上存在类似Gibbs的度量（Hα，V），我们证明\ [\ dim _h \ sigma（H _ {\ alpha，v}）= S_* ，\ Quad \ edline {\ dim} _b \，\ sigma（h _ {\ alpha，v}）= s^*，\ \]，其中s*和s*是下部和上部的预测）n≥1最终是周期性的，然后是s* = s*。
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 \[ \dim _H \sigma (H_{\alpha ,V})=s_*, \quad \overline {\dim }_B\, \sigma (H_{\alpha ,V})=s^*, \] where s* and s* are the lower and upper pre-dimensions. Moreover, if (an)n≥1 is ultimately periodic, then s* =s*.