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
MARMI, Stefano;
2010
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 $\alpha = 0$.File | Dimensione | Formato | |
---|---|---|---|
luzzi_marmi_nakada_natsui_jat_2010_finale.pdf
Accesso chiuso
Tipologia:
Altro materiale allegato
Licenza:
Non pubblico
Dimensione
943.68 kB
Formato
Adobe PDF
|
943.68 kB | Adobe PDF | Richiedi una copia |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.