arXiv math/06126022006-12-20math.DSmath.NT

随机$β$ -展开的不变密度

Invariant densities for random $β$-expansions

K. Dajani, M. de Vries

发表信息：: J. Eur. Math. Soc. 7 (2007), 157--176

摘要：

设$β>1$为非整数。我们考虑$\sum_{i=1}^{\infty} d_i β^{-i}$形式的扩展，其中数字$(d_i)_{i \geq 1}$通过定义在$\{0,1\}^{\N}\times [ 0, \lfloor β\rfloor /(β-1)]$上的Borel映射$K_β$生成。我们证明了一个绝对连续的$K_β$不变概率测度w.r.t.的存在性和唯一性。$m_p \otimes λ$，其中$m_p$是$\{0,1\}^{\N}$上的伯努利测度，参数为$p$ $(0 < p < 1)$，$λ$是$[0 ,\lfloor β\rfloor /(β-1)]$上的归一化勒贝格测度。此外，该度量具有$m_p \otimes μ_{β,p}$的形式，其中$μ_{β,p}$与$λ$等价。证明了极大熵测度和$m_p \otimes λ$是相互奇异的.在数字1具有正系数的有限贪婪展开的情况下，测度$m_p \otimes μ_{β,p}$是马尔可夫的。在最后一节中，我们回答了一个问题，关于一些普遍的扩张，一个概念介绍了[EK]。

英文摘要：

Let $β>1$ be a non-integer. We consider expansions of the form $\sum_{i=1}^{\infty} d_i β^{-i}$, where the digits $(d_i)_{i \geq 1}$ are generated by means of a Borel map $K_β$ defined on $\{0,1\}^{\N}\times [ 0, \lfloor β\rfloor /(β-1)]$. We show existence and uniqueness of an absolutely continuous $K_β$-invariant probability measure w.r.t. $m_p \otimes λ$, where $m_p$ is the Bernoulli measure on $\{0,1\}^{\N}$ with parameter $p$ $(0 < p < 1)$ and $λ$ is the normalized Lebesgue measure on $[0 ,\lfloor β\rfloor /(β-1)]$. Furthermore, this measure is of the form $m_p \otimes μ_{β,p}$, where $μ_{β,p}$ is equivalent with $λ$. We establish the fact that the measure of maximal entropy and $m_p \otimes λ$ are mutually singular. In case the number 1 has a finite greedy expansion with positive coefficients, the measure $m_p \otimes μ_{β,p}$ is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK].

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