On the metric theory of \(p\)-adic continued fractions
Let \(X=p\mathbb{Z}_p\) be the unit ball in the field of \(p\)-adic numbers \(\mathbb{Q}_p\), and define the map \(T:X\to X\) by \(T(x)=p^a/x-b\), where \(a=a(x)=v_p(x)\in\mathbb{N}\) is the \(p\)-adic valuation of \(x\), and \(b=b(x)\) is the unique element in \(\{1,\ldots,p-1\}\) such that \(p^a/x\equiv b \pmod{p\mathbb{Z}_p}\). The map \(T\) -- introduced by \textit{Th. Schneider} [Sympos. Math., Roma 4, 181--189 (1970; Zbl 0222.10035)] -- plays the rôle for \(p\)-adic continued fractions of the Gauss map for ordinary continued fractions. Indeed, writing \((b_n,a_n)\) for \(\bigl(b(T^{n-1}x),a(T^{n-1}x)\bigr)\), the symbolic sequence \(\{(b_n,a_n)\}\) determines \(x\) via \[ x= \cfrac{ p^{a_1}}{ b_1+\cfrac{ p^{a_2}}{ b_2+\cfrac{ p^{a_3}}{ \cdots}}} \] It is known [\textit{J. Hirsh} and \textit{L. C. Washington}, Ramanujan J. 25, No. 3, 389--403 (2011; Zbl 1234.11089)] that the dynamical system \((X,T,\mu)\) is ergodic w.r.t.~the Haar measure \(\mu\), normalized on \(X\). In this paper it is proved that \((X,T,\mu)\) is actually exact (i.e., the tail \(\sigma\)-algebra \(\bigcap_{n\geq 0}T^{-n}(\text{}\mu\)-measurable sets
scientific article; zbMATH DE number 6121458