Shift operators and two applications to \(\mathbb{F}_qT\)
Let \(K\) be a non-Archimedean local field, and \(O\) its ring of integers. Any \(x\in O\) has a canonical representation \(x=\sum\limits_{i=0}^\infty x_i\pi^i\) where \(\pi\) is a prime element, \(x_i\) are some representatives of the residue field. The \(\pi\)-adic shift \(S:\;O\to O\) is defined as \(S(x)=\frac1{\pi}(x-x_0)\). The author shows how to use the shift operator to obtain new and known orthonormal (in the non-Archimedean sense) bases for the space of \(\mathbb F_q\)-linear continuous functions on \(\mathbb F_qT\). By the digit principle \textit{K. Conrad} [J. Number Theory 84, No. 2, 230--257 (2000; Zbl 1017.11061)], this leads to the construction of bases in the space of all continuous functions. A similar approach is developed for \(O=\mathbb Z_p\). In the second part of the paper, the author obtains necessary and sufficient conditions for ergodicity of 1-Lipschitz functions on \(\mathbb F_2T\) represented by digit shift operators. Recently similar results were obtained for representations by Carlitz polynomials and digit derivatives \textit{D. Lin} et al. [Finite Fields Appl. 18, No. 3, 473--491 (2012; Zbl 1279.37051)]; \textit{S. Jeong} [J. Number Theory 133, No. 6, 1846--1863 (2013; Zbl 1263.37090)].
scientific article; zbMATH DE number 6308465
scientific article