On the average growth exponent for beta-expansions

TL;DR

This paper investigates the growth rate of $eta$-expansions for $eta$ in (1,2), showing ergodic behavior for generic points and exponential growth for certain $eta$, with explicit calculations at the golden ratio.

Contribution

It establishes the ergodic nature of the growth rate of $eta$-expansions for generic points and provides explicit growth exponents for specific $eta$, including the golden ratio.

Findings

Generic points have a uniform growth rate for their $eta$-expansions.

For $eta<rac{1+ oot{2} olinebreak 5}{2}$, the number of $eta$-expansions grows exponentially.

Explicit average growth exponent computed at the golden ratio, with applications to Bernoulli convolutions.

Abstract

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 $x$.