For $0\le\alpha\le 1$ given, we consider the one-parameter family of $\alpha$-continued fraction maps, which includes the Gauss map ($\alpha =1$), the nearest integer ($\alpha =1/2$) and by-excess ($\alpha =10$) continued fraction maps. To each of these expansions and to each choice of a positive function $u$ on the interval $I_\alpha$ we associate a generalized Brjuno function $B_{(\alpha , u)}(x)$. When $\alpha =1/2$ or $\alpha =1$, and $u(x)=-\log (x)$, these functions were introduced by Yoccoz in his work on linearization of holomorphic maps. We compare the functions obtained with different values of $\alpha$ and we prove that the set of $(\alpha , u)$-Brjuno numbers does not depend on the choice of provided that $\alpha \not= 0$. We then consider the case $\alpha = 0$, $u(x)=-log (x)$ and we prove that $x$ is a Brjuno number (for $\alpha \not= 0$) if and only if both $x$ and $-x$ are Brjuno numbers for $\alpha = 0$.

### Generalized Brjuno functions associated to a-continued fractions

#### Abstract

For $0\le\alpha\le 1$ given, we consider the one-parameter family of $\alpha$-continued fraction maps, which includes the Gauss map ($\alpha =1$), the nearest integer ($\alpha =1/2$) and by-excess ($\alpha =10$) continued fraction maps. To each of these expansions and to each choice of a positive function $u$ on the interval $I_\alpha$ we associate a generalized Brjuno function $B_{(\alpha , u)}(x)$. When $\alpha =1/2$ or $\alpha =1$, and $u(x)=-\log (x)$, these functions were introduced by Yoccoz in his work on linearization of holomorphic maps. We compare the functions obtained with different values of $\alpha$ and we prove that the set of $(\alpha , u)$-Brjuno numbers does not depend on the choice of provided that $\alpha \not= 0$. We then consider the case $\alpha = 0$, $u(x)=-log (x)$ and we prove that $x$ is a Brjuno number (for $\alpha \not= 0$) if and only if both $x$ and $-x$ are Brjuno numbers for $\alpha = 0$.
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2010
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/7138
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