Serendipitous decompositions of higher-dimensional continued fractions
HTML articles powered by AMS MathViewer
- by Anton Lukyanenko and Joseph Vandehey;
- Conform. Geom. Dyn. 29 (2025), 57-89
- DOI: https://doi.org/10.1090/ecgd/396
- Published electronically: January 21, 2025
Abstract:
We prove a suite of dynamical results, including exactness of the transformation and piecewise-analyticity of the invariant measure, for a family of continued fraction systems, including specific examples over reals, complex numbers, quaternions, octonions, and in $\mathbb {R}^3$. Our methods expand on the work of Nakada and Hensley, and in particular fill some gaps in Hensley’s analysis of Hurwitz complex continued fractions. We further introduce a new “serendipity” condition for a continued fraction algorithm, which controls the long-term behavior of the boundary of the fundamental domain under iteration of the continued fraction map, and which is under reasonable conditions equivalent to the finite range property. We also show that the finite range condition is extremely delicate: perturbations of serendipitous systems by non-quadratic irrationals do not remain serendipitous, and experimental evidence suggests that serendipity may fail even for some rational perturbations.References
- Pierre Arnoux and Thomas A. Schmidt, Cross sections for geodesic flows and $\alpha$-continued fractions, Nonlinearity 26 (2013), no. 3, 711–726. MR 3018939, DOI 10.1088/0951-7715/26/3/711
- Oscar F. Bandtlow and Oliver Jenkinson, Invariant measures for real analytic expanding maps, J. Lond. Math. Soc. (2) 75 (2007), no. 2, 343–368. MR 2340232, DOI 10.1112/jlms/jdm001
- Adriana Berechet, A Kuzmin-type theorem with exponential convergence for a class of fibred systems, Ergodic Theory Dynam. Systems 21 (2001), no. 3, 673–688. MR 1836426, DOI 10.1017/S014338570100133X
- Jérôme Chaubert, Minimum euclidien des ordres maximaux dans les algèbres centrales à division, 2007.
- J. H. Conway and N. J. A. Sloane, Voronoĭ regions of lattices, second moments of polytopes, and quantization, IEEE Trans. Inform. Theory 28 (1982), no. 2, 211–226. MR 651816, DOI 10.1109/TIT.1982.1056483
- John H. Conway and Derek A. Smith, On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters, Ltd., Natick, MA, 2003. MR 1957212
- H. S. M. Coxeter, Regular polytopes, 3rd ed., Dover Publications, Inc., New York, 1973. MR 370327
- Karma Dajani and Cor Kraaikamp, Ergodic theory of numbers, Carus Mathematical Monographs, vol. 29, Mathematical Association of America, Washington, DC, 2002. MR 1917322
- Hiromi Ei, Shunji Ito, Hitoshi Nakada, and Rie Natsui, On the construction of the natural extension of the Hurwitz complex continued fraction map, Monatsh. Math. 188 (2019), no. 1, 37–86. MR 3895391, DOI 10.1007/s00605-018-1229-0
- Manfred Einsiedler and Thomas Ward, Ergodic theory with a view towards number theory, Graduate Texts in Mathematics, vol. 259, Springer-Verlag London, Ltd., London, 2011. MR 2723325, DOI 10.1007/978-0-85729-021-2
- Robert W. Fitzgerald, Norm Euclidean quaternionic orders, Integers 12 (2012), no. 2, 197–208. MR 2955676, DOI 10.1515/integ.2011.096
- Doug Hensley, Continued fractions, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. MR 2351741, DOI 10.1142/9789812774682
- Ghaith Hiary and Joseph Vandehey, Calculations of the invariant measure for Hurwitz continued fractions, Exp. Math. 31 (2022), no. 1, 324–336. MR 4399128, DOI 10.1080/10586458.2019.1627255
- A. Hurwitz, Über die Entwicklung complexer Grössen in Kettenbrüche, Acta Math. 11 (1887), no. 1-4, 187–200 (German). MR 1554754, DOI 10.1007/BF02418048
- Marius Iosifescu and Cor Kraaikamp, Metrical theory of continued fractions, Mathematics and its Applications, vol. 547, Kluwer Academic Publishers, Dordrecht, 2002. MR 1960327, DOI 10.1007/978-94-015-9940-5
- Shunji Ito and Michiko Yuri, Number theoretical transformations with finite range structure and their ergodic properties, Tokyo J. Math. 10 (1987), no. 1, 1–32. MR 899469, DOI 10.3836/tjm/1270141789
- Svetlana Katok and Ilie Ugarcovici, Structure of attractors for $(a,b)$-continued fraction transformations, J. Mod. Dyn. 4 (2010), no. 4, 637–691. MR 2753948, DOI 10.3934/jmd.2010.4.637
- Cor Kraaikamp, A new class of continued fraction expansions, Acta Arith. 57 (1991), no. 1, 1–39. MR 1093246, DOI 10.4064/aa-57-1-1-39
- M. A. Krasnosel′skiĭ, Positive solutions of operator equations, P. Noordhoff Ltd., Groningen, 1964. Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron. MR 181881
- Anton Lukyanenko and Joseph Vandehey, Continued fractions on the Heisenberg group, Acta Arith. 167 (2015), no. 1, 19–42. MR 3310489, DOI 10.4064/aa167-1-2
- Anton Lukyanenko and Joseph Vandehey, Ergodicity of Iwasawa continued fractions via markable hyperbolic geodesics, Ergodic Theory Dynam. Systems 43 (2023), no. 5, 1666–1711. MR 4574152, DOI 10.1017/etds.2022.18
- Dieter H. Mayer, Approach to equilibrium for locally expanding maps in $\textbf {R}^{k}$, Comm. Math. Phys. 95 (1984), no. 1, 1–15. MR 757051
- Carminda Margaretha Mennen, The algebra and geometry of continued fractions with integer quaternion coefficients, Master’s Thesis, University of the Witwatersrand, 2015.
- Hitoshi Nakada, On the Kuzmin’s theorem for the complex continued fractions, Keio Engrg. Rep. 29 (1976), no. 9, 93–108. MR 463401
- 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
- Hitoshi Nakada and Rie Natsui, On the metrical theory of continued fraction mixing fibred systems and its application to Jacobi-Perron algorithm, Monatsh. Math. 138 (2003), no. 4, 267–288. MR 1973926, DOI 10.1007/s00605-002-0473-4
- Raghavan Narasimhan, Several complex variables, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1971. MR 342725
- Hans Peter Rehm, Prime factorization of integral Cayley octaves, Ann. Fac. Sci. Toulouse Math. (6) 2 (1993), no. 2, 271–289 (English, with English and French summaries). MR 1253392
- A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493. MR 97374, DOI 10.1007/BF02020331
- F. Schweiger and M. Waterman, Some remarks on Kuzmin’s theorem for $F$-expansions, J. Number Theory 5 (1973), 123–131. MR 319905, DOI 10.1016/0022-314X(73)90065-6
- F. Schweiger, Kuzmin’s theorem revisited, Ergodic Theory Dynam. Systems 20 (2000), no. 2, 557–565. MR 1756986, DOI 10.1017/S0143385700000286
- Shigeru Tanaka and Shunji Ito, On a family of continued-fraction transformations and their ergodic properties, Tokyo J. Math. 4 (1981), no. 1, 153–175. MR 625125, DOI 10.3836/tjm/1270215745
- Michael S. Waterman, A Kuzmin theorem for a class of number theoretic endomorphisms, Acta Arith. 19 (1971), 31–41. MR 288091, DOI 10.4064/aa-19-1-31-41
- Yoji Yoshii, Gausenstein integers, Toyama Math. J. 39 (2017), 9–18. MR 3792992
- Roland Zweimüller, Kuzmin, coupling, cones, and exponential mixing, Forum Math. 16 (2004), no. 3, 447–457. MR 2054836, DOI 10.1515/form.2004.021
Bibliographic Information
- Anton Lukyanenko
- Affiliation: Department of Mathematics, George Mason University, Fairfax, Virginia 22030
- MR Author ID: 995877
- ORCID: 0000-0002-2252-7407
- Email: alukyane@gmu.edu
- Joseph Vandehey
- Affiliation: Department of Mathematics, University of Texas at Tyler, Tyler, Texas 75799
- MR Author ID: 999182
- Email: jvandehey@uttyler.edu
- Received by editor(s): September 26, 2023
- Received by editor(s) in revised form: July 12, 2024, and September 5, 2024
- Published electronically: January 21, 2025
- © Copyright 2025 by the authors
- Journal: Conform. Geom. Dyn. 29 (2025), 57-89
- MSC (2020): Primary 11K50, 37A44; Secondary 11R52
- DOI: https://doi.org/10.1090/ecgd/396