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Renewal-type limit theorem for continued fractions with even partial quotients

Published online by Cambridge University Press:  03 February 2009

FRANCESCO CELLAROSI
FRANCESCO CELLAROSI*
Affiliation:
Mathematics Department, Princeton University, Princeton, NJ 08544-1000, USA (email: fcellaro@math.princeton.edu)
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Abstract

We prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by Schweiger [Continued fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg4 (1982), 59–70; On the approximation by continues fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg1–2 (1984), 105–114] and studied also by Kraaikamp and Lopes [The theta group and the continued fraction expansion with even partial quotients. Geom. Dedicata59(3) (1996), 293–333]. Our main result is proven following the strategy used by Sinai and Ulcigrai [Renewal-type limit theorem for the Gauss map and continued fractions. Ergod. Th. & Dynam. Sys.28 (2008), 643–655] in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 5 , October 2009 , pp. 1451 - 1478
Copyright
Copyright © Cambridge University Press 2009

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