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A two-dimensional singular function via Sturmian words in base \(\beta\). (English) Zbl 1297.11087

Summary: For \(\beta>1\), a function \((\cdot)_{\beta}\) maps each infinite word \(a_{1}a_{2}\cdots\in \mathbb N^{\mathbb N}\) to a real number \(\sum_{i=1}^\infty a_i/\beta^i\). We define \(\Xi(\alpha,\beta)\) by \((s'_{\alpha,0})_{\beta}\) where \(s'_{\alpha,0}\) is a lexicographically greatest mechanical word of slope \(\alpha\). This paper demonstrates that the function \(\Xi\) enjoys devil’s staircase-like properties. Its continuity, partial and total differentiability will be investigated. We also present a set of \(\Xi\)-values, in which any finite number of members are algebraically independent over the field of rationals.

MSC:

11J83 Metric theory
26A30 Singular functions, Cantor functions, functions with other special properties
37B10 Symbolic dynamics
68R15 Combinatorics on words
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