Metric properties and exceptional sets of \(\beta \)-expansions over formal Laurent series
Let \(\mathbb{F}\) be a finite field with \(q\) elements, and consider the field of formal Laurent series \(\mathbb{F}((X^{-1}))\). This is a complete metric space under the norm \(\| x\|=q^{-v(x)}\), where \(v\) is the standard exponential valuation. Let \(I=\{x:\| x\|<1\}\); as an abelian group, we have the direct sum decomposition \(\mathbb{F}((X^{-1}))=\mathbb{F}[X]\oplus I\), with projection functions \([-]\) and \(\{-\}\) onto the first and second summand, respectively. Fix now \(\beta\in\mathbb{F}((X^{-1}))\) with \(\|\beta\|>1\), and define the \(\beta\)-transformation \(T_\beta:I\to I\) by \(T_\beta(x)=\{\beta x\}\). The mapping \(T_\beta\) is a direct analogue of the well known \(\beta\)-transformation on the real unit interval \(R_\beta:[0,1)\to[0,1)\); here \(\beta>1\) is a fixed irrational number, and \(R_\beta(x)\) is the fractional part of \(\beta x\). The transformations \(T_\beta\) and \(R_\beta\) have obvious analogies, but also remarkable differences, the Laurent series case being somehow smoother. For example, the sets of all \(x\in I\) whose \(T_\beta\)-orbit is finite, or eventually periodic, have been characterized by \textit{M. Hbaib} and \textit{M. Mkaouar} [Int. J. Number Theory 2, No. 3, 365--378 (2006; Zbl 1157.11004)] and by \textit{K. Scheicher} [Finite Fields Appl. 13, No. 2, 394--410 (2007; Zbl 1152.11037)], while the real case is not yet fully understood. Also, it is known that not every sequence \(\varepsilon\in\{0,1,\dots,\lfloor\beta\rfloor\}^\mathbb{N}\) is the \(R_\beta\)-expansion of some \(x\in[0,1)\) (\(\varepsilon\) is the \(R_\beta\)-expansion of \(x\) if \(\varepsilon_n=\lfloor \beta R_\beta^{n-1}(x) \rfloor\), for every \(n\geq1\)). On the other hand, every sequence in \(\{\alpha\in\mathbb{F}[X]:\|\alpha\|<\|\beta\|\}^\mathbb{N}\) is the \(T_\beta\)-expansion of some \(x\in I\). In the present paper it is proved that the Haar measure on the compact abelian group \(I\) is \(T_\beta\)-invariant, and that the resulting dynamical system is ergodic. Some consequences of this fact are listed, such as the i.i.d.\ property of certain related stochastic processes, the explicit value of the metric entropy, the law of large numbers, the central limit theorem, the law of iterated logarithm. Also, the Hausdorff dimensions of certain exceptional sets are determined.
scientific article; zbMATH DE number 5373508
scientific article