Metrical diophantine approximation for continued fraction like maps of the interval

Haas, Andrew; Molnar, David

doi:10.1090/S0002-9947-03-03371-3

Metrical diophantine approximation for continued fraction like maps of the interval
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by Andrew Haas and David Molnar

Trans. Amer. Math. Soc. 356 (2004), 2851-2870

DOI: https://doi.org/10.1090/S0002-9947-03-03371-3

Published electronically: July 24, 2003

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Abstract:

We study the metrical properties of a class of continued fraction-like mappings of the unit interval, each of which is defined as the fractional part of a Möbius transformation taking the endpoints of the interval to zero and infinity.

References

Roy Adler and Leopold Flatto, Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 229–334. MR 1085823, DOI 10.1090/S0273-0979-1991-16076-3
Tim Bedford, Michael Keane, and Caroline Series (eds.), Ergodic theory, symbolic dynamics, and hyperbolic spaces, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991. Papers from the Workshop on Hyperbolic Geometry and Ergodic Theory held in Trieste, April 17–28, 1989. MR 1130170
Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0192027
Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
W. Bosma, H. Jager, and F. Wiedijk, Some metrical observations on the approximation by continued fractions, Nederl. Akad. Wetensch. Indag. Math. 45 (1983), no. 3, 281–299. MR 718069, DOI 10.1016/1385-7258(83)90063-X
Robert M. Burton, Cornelis Kraaikamp, and Thomas A. Schmidt, Natural extensions for the Rosen fractions, Trans. Amer. Math. Soc. 352 (2000), no. 3, 1277–1298. MR 1650073, DOI 10.1090/S0002-9947-99-02442-3
Harvey Cohn, Representation of Markoff’s binary quadratic forms by geodesics on a perforated torus, Acta Arith. 18 (1971), 125–136. MR 288079, DOI 10.4064/aa-18-1-125-136
Caroline Series, The modular surface and continued fractions, J. London Math. Soc. (2) 31 (1985), no. 1, 69–80. MR 810563, DOI 10.1112/jlms/s2-31.1.69
I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR 832433, DOI 10.1007/978-1-4615-6927-5
Karlheinz Gröchenig and Andrew Haas, Backward continued fractions, Hecke groups and invariant measures for transformations of the interval, Ergodic Theory Dynam. Systems 16 (1996), no. 6, 1241–1274. MR 1424398, DOI 10.1017/S0143385700010014
Andrew Haas, Invariant measures and natural extensions, Canad. Math. Bull. 45 (2002), no. 1, 97–108. MR 1884139, DOI 10.4153/CMB-2002-011-4
H. Jager, The distribution of certain sequences connected with the continued fraction, Nederl. Akad. Wetensch. Indag. Math. 48 (1986), no. 1, 61–69. MR 834320, DOI 10.1016/1385-7258(86)90006-5
A. Ya. Khintchine, Continued fractions, P. Noordhoff Ltd., Groningen, 1963. Translated by Peter Wynn. MR 0161834
Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8, Springer-Verlag, Berlin, 1987. Translated from the Portuguese by Silvio Levy. MR 889254, DOI 10.1007/978-3-642-70335-5
Hitoshi Nakada, Metrical theory for a class of continued fraction transformations and their natural extensions, Tokyo J. Math. 4 (1981), no. 2, 399–426. MR 646050, DOI 10.3836/tjm/1270215165
G. J. Rieger, Mischung und Ergodizität bei Kettenbrüchen nach nächsten Ganzen, J. Reine Angew. Math. 310 (1979), 171–181 (German). MR 546670, DOI 10.1515/crll.1979.310.171
Alfréd Rényi, On algorithms for the generation of real numbers, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 7 (1957), 265–293 (Hungarian). MR 0097645
Andrew M. Rockett and Peter Szüsz, Continued fractions, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. MR 1188878, DOI 10.1142/1725
V. A. Rohlin, Exact endomorphism of a Lebesgue space, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 14 (1964), 443–474 (Hungarian). MR 0228654
Fritz Schweiger, Ergodic theory of fibred systems and metric number theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. MR 1419320
F. Schweiger, Multidimensional Continued Fractions. Oxford Univ. Press, 2000.
S. M. Rudolfer and K. M. Wilkinson, A number-theoretic class of weak Bernoulli transformations, Math. Systems Theory 7 (1973), 14–24. MR 323744, DOI 10.1007/BF01824802