On a Gauss-Kuzmin-Type Problem for a Family of Continued Fraction Expansions

Abstract

In this paper we study in detail a family of continued fraction expansions of any number in the unit closed interval $[0,1]$ whose digits are differences of consecutive non-positive integer powers of an integer $m\ge 2$. For the transformation which generates this expansion and its invariant measure, the Perron-Frobenius operator is given and studied. For this expansion, we apply the method of random systems with complete connections by Iosifescu and obtained the solution of its Gauss-Kuzmin type problem.

This paper has been withdrawn by the author due to a change of the team of authors

Cited by in corpus (8)

• A gauss-kuzmin theorem and related questions for $\theta$-expansions
• On certain generalizations of one function and related problems (Systems of functional equations and generalizations of certain functions)
• Dependence with complete connections and the Gauss-Kuzmin theorem for N-continued fractions
• A Gauss-Kuzmin theorem for continued fractions associated with non-positive interger powers of an integer $m\ge 2$
• Metric properties of N-continued fractions
• A dependence with complete connections approach to generalized Rényi continued fractions
• Ergodic Properties Of $\theta$-Expansions And A Gauss-Kuzmin-Type Problem
• A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms