Continued fractions and Brjuno functions
The regular continued fraction expansion of a positive real number \(x\) is given by \[ x=b_0+{1\over b_1} {\;\atop +} {1\over b_2} {\;\atop +\dots} , \tag{1} \] where the elements \(b_n\) are derived by the following recursive procedure: (2) \(x_0:= x\), \(b_n:=[x_n]\) (i.e., \(b_n\in \mathbb N_0:=\mathbb N\cup\{0\}\) and \(0\leq x_n-b_n<1\), \(x_{n+1}:={1\over x_n-b_n}\) for \(n=0,1,2,\dots\;\). The continued fraction terminates (i.e., \(x_n=\infty\) for some \(n\in\mathbb N\)) if and only if \(x\) is a rational number, and it always converges to \(x\). Nakada has generalized this idea to something that is sometimes called a Japanese continued fraction expansion of \(x\): Let \(\alpha\) be a fixed number, \(0\leq\alpha\leq 1\). The recursive procedure (2) is then modified to (3) \(x_0:=x\), \(a_n:=[x_n]_\alpha:=[x-\alpha+1]\) (i.e., \(a_n\in\mathbb N_0\) and \(-1+\alpha\leq x_n-a_n<\alpha)\), \(\varepsilon_{n+1}x_{n+1}:={1\over x_n-a_n}\), where \(x_{n+1}\geq 0\) and \(\varepsilon_{n+1}=\pm 1\), for \(n=0,1,2,\dots\). This gives the continued fraction \[ x=a_0+{\varepsilon_1\over a_1} {\;\atop +} {\varepsilon_2\over a_2} {\;\atop +\dots} , \tag{4} \] which always converges to \(x\) and terminates if and only if \(x\) is a rational number, unless \(\alpha=0\) which is a somewhat special case. Associated to (4) the authors introduce the Brjuno function \[ B_\alpha (x): =\sum_{n=0}^\infty \beta_{n-1}\ln(x_n^{-1})\in\widehat{\mathbb C}:=\mathbb C\cup \{\infty\},\quad \text{where }\quad \beta_n:=x_0x_1\cdots x_n \] which is set equal to \(\infty\) if (4) terminates. \(x\) is called a Brjuno number if the series (of positive numbers) converges to a finite value. \(B_\alpha (x)\) reduces to the Brjuno functions introduced by Yoccoz for the special cases \(\alpha={1\over 2}\) and \(\alpha=1\). The authors prove a number of properties for (4) and \(B_\alpha(x)\) and compare them to earlier results for (1) and \(B_1(x)\).
scientific article; zbMATH DE number 1334882