On the average growth exponent for beta-expansions
| dc.creator | Sidorov, Nikita | |
| dc.date | 2008-08-11 | |
| dc.date | 2008-09-24 | |
| dc.date.accessioned | 2026-07-07T10:04:50Z | |
| dc.date.available | 2026-07-07T10:04:50Z | |
| dc.description | Let $\be\in(1,2)$. Each $x\in I_\be:=[0,\frac{1}{\be-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty a_k\be^{-k}, \] where $a_k\in\{0,1\}$ for all $k$ (a $\be$-expansion of $x$). It was shown in \cite{S} that a.e. $x\in I_\be$ has a continuum of distinct $\be$-expansions. In this paper we show that for a generic $x$, this continuum has one and the same growth rate, i.e., the general $\be$-expansions exhibit an ergodic behaviour. When $\be<\frac{1+\sqrt5}2$, we show that the set of $\be$-expansions grows exponentially for every $x\in(0,\frac{1}{\be-1})$. Special attention is paid to the case $\be=\frac{1+\sqrt5}2$, for which we explicitly compute the average growth exponent and apply this result to evaluating the local dimension of the corresponding Bernoulli convolution at a Lebesgue-generic $x$. | |
| dc.description | This paper has been withdrawn by the author, due a crucial error in the proof of Theorem 2.3 | |
| dc.identifier | https://arxiv.org/abs/0808.1589 | |
| dc.identifier | http://arxiv.org/abs/0808.1589 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/169824 | |
| dc.subject | Dynamical Systems | |
| dc.subject | Number Theory | |
| dc.subject | 11A63; 28D05; 42A85 | |
| dc.title | On the average growth exponent for beta-expansions | |
| dc.type | text |