The recurrence set arising in 伪-L眉roth transformation

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• 作者：Luming Shen ; ^{lum_s@126.com" class="auth_mail" title="E-mail the corresponding author} ; Xinqiang Li
• 关键词：11J70 ; 28A80
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：159
• 期：Complete
• 页码：76-88
• 全文大小：307 K

文摘

Let X1500253X&_mathId=si1.gif&_user=111111111&_pii=S0022314X1500253X&_rdoc=1&_issn=0022314X&md5=fea71ab905ff880bdd1fb861d4a09b3d" title="Click to view the MathML source">伪={A_n}_n≥1$伪={\{A_n\}}_{n\ge 1}$ be a sequence of left-open and right-closed intervals which partition (0,1]$(0,1]$. The 伪  -Lüroth transformation L_伪$L_{伪}$ is defined as an infinite piecewise linear map which maps A_n$A_n$ linearly onto (0,1]$(0,1]$ for every n≥1$n\ge 1$. Then every point x∈(0,1]$x\in (0,1]$ is attached with a finite or infinite integer sequence {鈩?sub>n}_n≥1${\{{\mathrm{鈩?/mi>}}_n\}}_{n\ge 1}$ by looking at the coding of its trajectory. In this note, we consider the size of the recurrent set in such a system. More precisely, let x₀∈(0,1]$x_0\in (0,1]$ with an infinite 伪  -Lüroth expansion, and {t_n}_n≥1${\{t_n\}}_{n\ge 1}$ an arbitrary non-decreasing sequence of natural numbers. The recurrence set of 伪  -Lüroth transformation L_伪$L_{伪}$ is defined as where A(鈩?sub>1,鈩?sub>2,鈰?鈩?sub>t_n)(x₀) denotes the t_n$t_n$-th cylinder containing 15ed2a42b63f41464f59b6434f" title="Click to view the MathML source">x₀$x_0$ in 伪  -Lüroth expansion. The Hausdorff dimension of F(x₀)$F(x_0)$ is obtained.