Natural extensions for the Rosen fractions

Authors:
Robert M. Burton, Cornelis Kraaikamp and Thomas A. Schmidt

Journal:
Trans. Amer. Math. Soc. **352** (2000), 1277-1298

MSC (2000):
Primary 11J70, 37A25

DOI:
https://doi.org/10.1090/S0002-9947-99-02442-3

Published electronically:
October 15, 1999

MathSciNet review:
1650073

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Abstract | References | Similar Articles | Additional Information

Abstract: The Rosen fractions form an infinite family which generalizes the nearest-integer continued fractions. We find planar natural extensions for the associated interval maps. This allows us to easily prove that the interval maps are weak Bernoulli, as well as to unify and generalize results of Diophantine approximation from the literature.

**[A]**R. Adler,*Continued fractions and Bernoulli trials*, in*Ergodic Theory (A Seminar)*, J. Moser, E. Phillips and S. Varadhan, eds., Courant Inst. of Math. Sci. (Lect. Notes 110), 1975, New York. MR**58:6177****[B]**A. Beardon,*The Geometry of Discrete Groups*, Graduate Texts in Mathematics**91**, Springer-Verlag 1983, New York. MR**85d:22026****[Ber1]**H.C.P. Berbee,*Random walks with stationary increments and renewal theory*, Math. Centre Tracts**112**, Mathematisch Centrum, Amsterdam 1979. MR**81e:60093****[Ber2]**H.C.P. Berbee,*Periodicity and absolute regularity*, Israel J. Math.**55**(1986), 289-304. MR**88b:60088****[GH]**K. Gröchenig and A. Haas,*Backward continued fractions and their invariant measures*, Canad. Math. Bull.**39**(1996), 186-198. MR**97h:11077****[HS]**A. Haas and C. Series,*Hurwitz constants and Diophantine approximation on Hecke groups*, J. London Math. Soc.**34**(1986), 219-234. MR**87m:11060****[JK]**H. Jager and C. Kraaikamp,*On the approximation by continued fractions*, Indag. Math.**51**(1989), 289-307. MR**90k:11084****[K1]**C. Kraaikamp,*The distribution of some sequences connected with the nearest integer continued fraction*, Indag. Math.**49**(1987), 177-191. MR**88j:11045****[K2]**C. Kraaikamp,*A new class of continued fraction expansions*, Acta Arith.**57**(1991), 1-39. MR**92a:11090****[KL]**C. Kraaikamp and A. Lopes,*The theta group and the continued fraction with even partial quotients*, Geom. Dedicata**59**(1996), 293-333. MR**97g:58135****[L1]**J. Lehner,*Diophantine approximation on Hecke groups*, Glasgow Math. J.**27**(1985), 117-127. MR**87e:11079****[L2]**J. Lehner,*The local Hurwitz constant and Diophantine approximation on Hecke groups*, Math. Comp.**55**(1990), 765-781. MR**91c:11036****[N1]**H. Nakada,*Metrical theory for a class of continued fraction transformations*, Tokyo J. Math.**4**(1981), 399-426. MR**83k:10095****[N2]**H. Nakada,*Continued fractions, geodesic flows and Ford circles*, in:*Algorithms, Fractals and Dynamics,*Ed. Y. Takahashi, Plenum Press, 1995, 179-191. MR**97g:11086****[NIT]**H. Nakada, Sh. Ito, S. Tanaka,*On the invariant measure for the transformations associated with some real continued-fractions*, Keio Engrg. Rep.**30**(1977), 159-175. MR**58:16574****[R]**G. J. Rieger,*Mischung und Ergodizität bei Kettenbrüchen nach nächsten Ganzen*, J. Reine Angew. Math.**310**(1979), 171-181. MR**81c:10066****[Ro]**D. Rosen,*A class of continued fractions associated with certain properly discontinuous groups*, Duke Math. J.**21**(1954), 549-563. MR**16:458d****[S]**T. A. Schmidt,*Remarks on the Rosen - continued fractions*, in*Number theory with an emphasis on the Markoff spectrum*, A. Pollington, W. Moran, eds., Dekker, New York, 1993, 227-238. MR**94i:11034****[W]**H. Weber,*Lehrbuch der Algebra, I*, 2nd edition, Viehweg, Braunschweig, 1898.**[Wo]**J. Wolfart,*Diskrete Deformation Fuchsscher Gruppen und ihrer automorphen Formen*, J. Reine Angew. Math.**348**(1984), 203-220. MR**85c:11044**

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Additional Information

**Robert M. Burton**

Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 97331

Email:
burton@math.orst.edu

**Cornelis Kraaikamp**

Affiliation:
Technische Universiteit Delft & Thomas Stieltjes Institute of Mathematics, ITS (SSOR), Mekelweg 4, 2628 CD Delft, the Netherlands

Email:
c.kraaikamp@its.tudelft.nl

**Thomas A. Schmidt**

Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 97331

Email:
toms@math.orst.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02442-3

Received by editor(s):
August 1, 1997

Published electronically:
October 15, 1999

Additional Notes:
The first author was partially supported by AFOSR grant 93-1-0275 and NSF grant DMS 96-26575. The third author was partially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)

Article copyright:
© Copyright 1999
American Mathematical Society