Algebraic and ergodic properties of a new continued fraction algorithm with non-decreasing partial quotients
Journal de Théorie des Nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 497-516.

In this paper the Engel continued fraction (ECF) expansion of any x(0,1) is introduced. Basic and ergodic properties of this expansion are studied. Also the relation between the ECF and F. Ryde’s monotonen, nicht-abnehmenden Kettenbruch (MNK) is studied.

On introduit la notion de développement en fractions continues de Engel. Nous étudions notamment les propriétés ergodiques de ce développement et le lien avec celui introduit par F. Ryde monotonen, nicht-abnehmenden Kettenbruch.

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     author = {Yusuf Hartono and Cor Kraaikamp and Fritz Schweiger},
     title = {Algebraic and ergodic properties of a new continued fraction algorithm with non-decreasing partial quotients},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
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     publisher = {Universit\'e Bordeaux I},
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Yusuf Hartono; Cor Kraaikamp; Fritz Schweiger. Algebraic and ergodic properties of a new continued fraction algorithm with non-decreasing partial quotients. Journal de Théorie des Nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 497-516. doi : 10.5802/jtnb.371. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.371/

[AF] R.L. Adler, L. Flatto, The backward continued fraction map and geodesic flow. Ergodic Theory Dynam. Systems 4 (1984), no. 4, 487-492. MR 86h:58116 | MR: 779707 | Zbl: 0563.58019

[B] E. Borel, Sur les développements unitaires normaux. C. R. Acad. Sci. Paris 225, (1947), 51. MR 9,292c | MR: 23007 | Zbl: 0029.15303

[DK] K. Dajani, C. Kraaikamp, The Mother of All Continued Fractions. Coll. Math. 84/85 (2000), 109-123. | MR: 1778844 | Zbl: 0961.11027

[ERS] P. Erdös, A. Rényi, P. Szüsz, On Engel's and Sylvester's series. Ann. Univ. Sci. Budapest. Etvs. Sect. Math. 1 (1958), 7-32. MR 21#1288 | MR: 102496 | Zbl: 0107.27002

[G] J. Galambos, Representations of real numbers by infinite series. Lecture Notes in Mathematics 502, Springer-Verlag, Berlin-New York, 1976. MR 58#27873 | MR: 568141 | Zbl: 0322.10002

[HL] P. Hubert, Y. Lacroix, Renormalization of algorithms in the probabilistic sense. New trends in probability and statistics, Vol. 4 (Palanga, 1996), 401-412, VSP, Utrecht, 1997. MR 2000c:11131 | MR: 1653625 | Zbl: 1040.11511

[K] C. Kraaikamp, A new class of continued fraction expansions. Acta Arith. 57 (1991), 1-39. MR 92a:11090 | MR: 1093246 | Zbl: 0721.11029

[K2K] S. Kalpazidou, A. Knopfmacher, J. Knopfmacher, Lüroth-type alternating series representations for real numbers. Acta Arith. 55 (1990), 311-322. MR 91i:11011 | MR: 1069185 | Zbl: 0702.11048

[Leh] J. Lehner, Semiregular continued fractions whose partial denominators are 1 or 2. Contemp. Math. 169 (1994), 407-410. MR 95e:11011 | MR: 1292915 | Zbl: 0814.11008

[L] P. Lévy, Remarques sur un théorème de M. Émile Borel. C. R. Acad. Sci. Paris 225 (1947), 918-919. MR 9,292d | MR: 23008 | Zbl: 0029.15304

[R] A. Rényi, A new approach to the theory of Engel's series. Ann. Univ. Sci. Budapest. Etvs Sect. Math. 5 (1962), 25-32. MR 27#126 | MR: 150123 | Zbl: 0232.10028

[Ry1] F. Ryde, Eine neue Art monotoner Kettenbruchentwicklungen. Ark. Mat. 1 (1951), 319-339. MR 13,115c | MR: 42457 | Zbl: 0042.29602

[Ry2] F. Ryde, Sur les fractions continues monotones nondécroissantes périodiques. Ark. Mat. 1 (1951), 409-420. MR 13,115d | MR: 42458 | Zbl: 0042.29603

[Si] W. Sierpinski, Sur quelques algorithmes pour développer les nombres réels en séries. In: Oeuvres choisies Tome I, Warszawa 1974, 236-254. MR 54#2405 | MR: 414302

[S1] F. Schweiger, Ergodische Theorie der Engelschen und Sylvesterschen Reihen. Czechoslovak Math. J. 20 (95) 1970, 243-245. MR 41#3712; Czechoslovak Math. J. 21 (96) 1971, 165. MR 43#2190 | MR: 259070 | Zbl: 0197.34501

[S2] F. Schweiger, Metrische Ergebnisse über den Kotangensalgorithmus. Acta Arith. 26 (1975), 217-222. MR 51#10269 | MR: 374069 | Zbl: 0261.10042

[S3] F. Schweiger, Ergodic theory of fibred systems and metric number theory. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. MR 97h:11083 | MR: 1419320 | Zbl: 0819.11027

[T] M. Thaler, σ-endliche invariante Masse für die Engelschen Reihen. Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 116 (1979), no. 2, 46-48. MR 80j:28028 | Zbl: 0412.10038

[V] W. Vervaat, Success epochs in Bernoulli trials (with applications in number theory). Mathematical Centre Tracts 42, Mathematisch Centrum, Amsterdam, 1972. MR 48#7331 | MR: 328989 | Zbl: 0267.60003

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