zbMATH — the first resource for mathematics

On a Gauss-Kuzmin type problem for piecewise fractional linear maps with explicit invariant measure. (English) Zbl 0971.60101
Consider the map \(T:[0,1]\to [0,1]\) defined as \[ T(x)=\begin{cases} x/\bigl(1-(N+1) x\bigr)\quad & \text{if }0\leq x\leq 1/(N+2),\\ (1-kx)/x \quad & \text{if }1/(k+1)<x\leq 1/k,\;1\leq k\leq N+1, \end{cases} \] where \(N\) is a fixed positive integer. According to F. Schweiger [J. Aust. Math. Soc., Ser. A 34, 55-59 (1983; Zbl 0514.28012)], the map \(T\) has an invariant measure whose density can be expressed in closed form. The author derives a Gauss-Kuzmin-Lévy type theorem, that is, the asymptotic behaviour of \(\mu((x:T^nx<y))\), \(y \in[0,1]\), as \(n\to\infty\), where \(\mu\) is a nonatomic probability measure on the Borel subsets of \([0,1]\). The author’s approach is via the theory of dependence with complete connections, see the reviewer and S. Grigorescu [“Dependence with complete connections and its applications” (1990; Zbl 0749.60067)], and is patterned after the work of S. Kalpazidou in similar contexts [Rev. Roum. Math. Pures Appl. 30, 527-537 (1985; Zbl 0576.60100), and Lith. Math. J. 27, No. 1, 32-40 (1987) and Litov. Mat. Sb. 27, No. 1, 68-79 (1987; Zbl 0644.10035)].

60K99 Special processes
28D05 Measure-preserving transformations
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Full Text: DOI EuDML