A Renyi s Backward Continued Fraction Map

Abstract

We introduce and study in detail a special class of backward continued fractions that represents a generalization of Rényi continued fractions. We investigate the main metrical properties of the digits occurring in these expansions and we construct the natural extension for the transformation that generates the Rényi-type expansion. We also define the random system with complete connections associated with the underlying dynamical system whose ergodic behaviour allows us to prove a variant of Gauss–Kuzmin-type theorem.

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Acknowledgement

The authors thank the referee for carefully reading our manuscript and for giving us such constructive comments. These helped us to improving the quality of the paper.

Author information

Authors and Affiliations

1. Mircea cel Batran Naval Academy, 1 Fulgerului, 900218, Constanta, Romania

   D. Lascu

2. Faculty of Applied Sciences, Politehnica University of Bucharest, Splaiul Independentei 313, 060042, Bucharest, Romania

   G. I. Sebe

3. Institute of Mathematical Statistics and Applied Mathematics, Calea 13 Sept. 13, 050711, Bucharest, Romania

   G. I. Sebe

Corresponding author

Correspondence to D. Lascu.

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Cite this article

Lascu, D., Sebe, G.I. A dependence with complete connections approach to generalized Rényi continued fractions. Acta Math. Hungar. 160, 292–313 (2020). https://doi.org/10.1007/s10474-019-00974-x

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• Received: 09 March 2019

• Revised: 15 May 2019

• Accepted: 16 May 2019

• Published: 04 September 2019

• Issue Date: April 2020

• DOI : https://doi.org/10.1007/s10474-019-00974-x

Key words and phrases

• Rényi continued fraction
• Perron–Frobenius operator
• random system with complete connections
• Gauss–Kuzmin problem

Mathematics Subject Classification

• 11J70
• 28D05
• 37A30
• 60A10

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