Mahler’s theorem says that a continuous function \(f\) from the ring \(\mathbb{Z}_p\) of \(p\)-adic integers to itself can be uniquely expanded as \(f(x)=\sum_{i=0}^{\infty} a_i {x\choose i}\) with \(a_i\in \mathbb{Z}_p\), and \(a_i\to 0\) as \(i\to \infty\). In 1994, by using the Mahler coefficients of \(f\), V. Anashin [Math. Notes 55, No. 2, 109–133 (1994); translation from Mat. Zametki 55, No. 2, 3–46 (1994; Zbl 0835.11031)] gave sufficient and necessary conditions for \(f\) to be \(1\)-Lipschitz or measure-preserving (with respect to the Haar measure). Furthermore, he obtained some sufficient conditions (also necessary for \(p=2\)) for \(f\) to be ergodic. The present paper generalizes the results of V. Anashin to continuous transformations on the ring \(\mathbb{F}_r[[T]]\) of formal power series over finite fields \(\mathbb{F}_r\). In place of the polynomials \({x\choose i}\) in the case of \(\mathbb{Z}_p\), the Carlitz polynomials form an orthonormal basis for the continuous functions space of \(\mathbb{F}_r[[T]]\) and a continuous function on \(\mathbb{F}_r[[T]]\) has a Carlitz expansion. The authors find sufficient and necessary conditions on the Carlitz coefficients for a continuous transformation on \(\mathbb{F}_r[[T]]\) to be \(1\)-Lipschitz. Like in the ergodic \(1\)-Lipschitz theory over \(\mathbb{Z}_2\) of V. Anashin, sufficient and necessary conditions on the Carlitz coefficients for the ergodicity of \(1\)-Lipschitz transformations on \(\mathbb{F}_2[[T]]\) are obtained.
Another important expansion of a continuous function on \(\mathbb{Z}_p\) is the Van der Put expansion using the Van der Put basis (characteristic functions of balls). Sufficient and necessary conditions on the Van der Put coefficients for a continuous transformation on \(\mathbb{Z}_p\) to be \(1\)-Lipschitz, or measure-preserving were given by V. Anashin, A. Khrennikov and E. Yurova [Dokl. Math. 83, No. 3, 306–308 (2011); translation from Dokl. Akad. Nauk 438, No. 2, 151–153 (2011; Zbl 1247.37007)] and A. Khrennikov and E. Yurova [J. Number Theory 133, No. 2, 484–491 (2013; Zbl 1278.37061)]. An ergodic criterion of a \(1\)-Lipschitz transformation on \(\mathbb{Z}_2\) using the Van der Put coefficients was obtained in [Anashin et al. loc. cit., Zbl 1247.37007]. The other main part of this article is devoted to finding the corresponding theory on \(\mathbb{F}_r[[T]]\). It characterizes the form of Van der Put expansion that a \(1\)-Lipschitz transformation on \(\mathbb{F}_r[[T]]\) should have. The criteria using the Van der Put coefficients for measure-preservation and ergodicity of a \(1\)-Lipschitz transformation on \(\mathbb{F}_2[[T]]\) are derived.
In the end of the paper, the authors comment that the ergodic theory (ergodic criteria) over \(\mathbb{F}_r[[T]]\) with \(r\) being a power of a prime number is left for future study. Very recently some sufficient conditions for the ergodicity of \(1\)-Lipschitz transformations on \(\mathbb{Z}_p\) for general prime \(p\) were obtained by S. Jeong [J. Number Theory 133, 2874–2891 (2013)].