On the second Lyapunov exponent of some multidimensional continued fraction algorithms
HTML articles powered by AMS MathViewer
- by Valérie Berthé, Wolfgang Steiner and Jörg M. Thuswaldner HTML | PDF
- Math. Comp. 90 (2021), 883-905 Request permission
Abstract:
We study the strong convergence of certain multidimensional continued fraction algorithms. In particular, in the two- and three-dimensional case, we prove that the second Lyapunov exponent of Selmer’s algorithm is negative and bound it away from zero. Moreover, we give heuristic results on several other continued fraction algorithms. Our results indicate that all classical multidimensional continued fraction algorithms cease to be strongly convergent for high dimensions. The only exception seems to be the Arnoux–Rauzy algorithm which, however, is defined only on a set of measure zero.References
- Apple Developer, Metal shading language specification, Technical Report Version 2.2, Cupertino CA, USA, 2019.
- Pierre Arnoux and Sébastien Labbé, On some symmetric multidimensional continued fraction algorithms, Ergodic Theory Dynam. Systems 38 (2018), no. 5, 1601–1626. MR 3819994, DOI 10.1017/etds.2016.112
- Pierre Arnoux and Štěpán Starosta, The Rauzy gasket, Further developments in fractals and related fields, Trends Math., Birkhäuser/Springer, New York, 2013, pp. 1–23. MR 3184185, DOI 10.1007/978-0-8176-8400-6_{1}
- Artur Avila and Vincent Delecroix, Some monoids of Pisot matrices, New trends in one-dimensional dynamics, Springer Proc. Math. Stat., vol. 285, Springer, Cham, [2019] ©2019, pp. 21–30. MR 4043208
- Artur Avila, Pascal Hubert, and Alexandra Skripchenko, On the Hausdorff dimension of the Rauzy gasket, Bull. Soc. Math. France 144 (2016), no. 3, 539–568 (English, with English and French summaries). MR 3558432, DOI 10.24033/bsmf.2722
- V. Baladi and A. Nogueira, Lyapunov exponents for non-classical multidimensional continued fraction algorithms, Nonlinearity 9 (1996), no. 6, 1529–1546. MR 1419459, DOI 10.1088/0951-7715/9/6/008
- P. R. Baldwin, A convergence exponent for multidimensional continued-fraction algorithms, J. Statist. Phys. 66 (1992), no. 5-6, 1507–1526. MR 1156412, DOI 10.1007/BF01054431
- P. R. Baldwin, A multidimensional continued fraction and some of its statistical properties, J. Statist. Phys. 66 (1992), no. 5-6, 1463–1505. MR 1156411, DOI 10.1007/BF01054430
- Leon Bernstein, The Jacobi-Perron algorithm—Its theory and application, Lecture Notes in Mathematics, Vol. 207, Springer-Verlag, Berlin-New York, 1971. MR 0285478
- Valérie Berthé, Loïck Lhote, and Brigitte Vallée, The Brun gcd algorithm in high dimensions is almost always subtractive, J. Symbolic Comput. 85 (2018), 72–107. MR 3707852, DOI 10.1016/j.jsc.2017.07.004
- A. Broise-Alamichel and Y. Guivarc’h, Exposants caractéristiques de l’algorithme de Jacobi-Perron et de la transformation associée, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 3, 565–686 (French, with English and French summaries). MR 1838461
- Henk Bruin, Robbert Fokkink, and Cor Kraaikamp, The convergence of the generalised Selmer algorithm, Israel J. Math. 209 (2015), no. 2, 803–823. MR 3430260, DOI 10.1007/s11856-015-1237-x
- Henk Bruin, Robbert Fokkink, and Cor Kraaikamp, Erratum to: “The convergence of the generalised Selmer algorithm” [ MR3430260], Israel J. Math. 231 (2019), no. 1, 505. MR 3960016, DOI 10.1007/s11856-019-1861-y
- V. Brun, En generalisation av kjedebrøken I, Skr. Vidensk.-Selsk. Christiana Math.-Nat. Kl. 6 (1919), 1–29.
- V. Brun, En generalisation av kjedebrøken II, Skr. Vidensk.-Selsk. Christiana Math.-Nat. Kl. 6 (1920), 1–24.
- Viggo Brun, Algorithmes euclidiens pour trois et quatre nombres, Treizième congrès des mathématiciens scandinaves, tenu à Helsinki 18-23 août 1957, Mercators Tryckeri, Helsinki, 1958, pp. 45–64 (French). MR 0111735
- J. Cassaigne, S. Labbé, and J. Leroy, A set of sequences of complexity $2n+1$, Combinatorics on words, Lecture Notes in Comput. Sci., vol. 10432, Springer, Cham, 2017, pp. 144–156.
- N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, vol. 1794, Springer-Verlag, Berlin, 2002. Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. MR 1970385, DOI 10.1007/b13861
- C. Fougeron and A. Skripchenko, Simplicity of spectrum for certain multidimensional continued fraction algorithms, 2019. arXiv:1904.13297, to appear in Monatsh. Math.
- T. Fujita, S. Ito, M. Keane, and M. Ohtsuki, On almost everywhere exponential convergence of the modified Jacobi-Perron algorithm: a corrected proof, Ergodic Theory Dynam. Systems 16 (1996), no. 6, 1345–1352. MR 1424403, DOI 10.1017/S0143385700010063
- Thomas Garrity, On periodic sequences for algebraic numbers, J. Number Theory 88 (2001), no. 1, 86–103. MR 1825992, DOI 10.1006/jnth.2000.2608
- D. Goldberg, What every computer scientist should know about floating-point arithmetic., ACM Comput. Surv. 23 (1991), no. 1, 5–48. corrigendum: ACM Computing Surveys 23(3): 413 (1991), comments: ACM Computing Surveys 24(2): 319 (1992).
- D. M. Hardcastle, The three-dimensional Gauss algorithm is strongly convergent almost everywhere, Experiment. Math. 11 (2002), no. 1, 131–141. MR 1960307
- D. M. Hardcastle and K. Khanin, On almost everywhere strong convergence of multi-dimensional continued fraction algorithms, Ergodic Theory Dynam. Systems 20 (2000), no. 6, 1711–1733. MR 1804954, DOI 10.1017/S014338570000095X
- D. M. Hardcastle and K. Khanin, The $d$-dimensional Gauss transformation: strong convergence and Lyapunov exponents, Experiment. Math. 11 (2002), no. 1, 119–129. MR 1960306
- A. Herrera Torres, Simplicity of the Lyapunov spectrum of multidimensional continued fraction algorithms, 2009. PhD thesis, IMPA.
- S. Ito, M. Keane, and M. Ohtsuki, Almost everywhere exponential convergence of the modified Jacobi-Perron algorithm, Ergodic Theory Dynam. Systems 13 (1993), no. 2, 319–334. MR 1235475, DOI 10.1017/S0143385700007380
- Russell A. Johnson, Kenneth J. Palmer, and George R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal. 18 (1987), no. 1, 1–33. MR 871817, DOI 10.1137/0518001
- J. Christopher Kops, Selmer’s multiplicative algorithm, Integers 12 (2012), no. 1, 1–20. MR 2955578, DOI 10.1515/integ.2011.080
- S. Labbé, 3-dimensional continued fraction algorithms cheat sheets, 2015. arXiv:1511.08399.
- J. C. Lagarias, The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms, Monatsh. Math. 115 (1993), no. 4, 299–328. MR 1230366, DOI 10.1007/BF01667310
- Ronald Meester, A simple proof of the exponential convergence of the modified Jacobi-Perron algorithm, Ergodic Theory Dynam. Systems 19 (1999), no. 4, 1077–1083. MR 1709431, DOI 10.1017/S0143385799133960
- Ali Messaoudi, Arnaldo Nogueira, and Fritz Schweiger, Ergodic properties of triangle partitions, Monatsh. Math. 157 (2009), no. 3, 283–299. MR 2520730, DOI 10.1007/s00605-008-0065-z
- Kentaro Nakaishi, Exponentially strong convergence of non-classical multidimensional continued fraction algorithms, Stoch. Dyn. 2 (2002), no. 4, 563–586. MR 1949301, DOI 10.1142/S0219493702000546
- Kentaro Nakaishi, Strong convergence of additive multidimensional continued fraction algorithms, Acta Arith. 121 (2006), no. 1, 1–19. MR 2216301, DOI 10.4064/aa121-1-1
- R. E. A. C. Paley and H. D. Ursell, Continued fractions in several dimensions, Math. Proc. Cambridge Philos. Soc. 26 (1930), 127–144.
- E. V. Podsypanin, A generalization of the continued fraction algorithm that is related to the Viggo Brun algorithm, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 67 (1977), 184–194, 227 (Russian). Studies in number theory (LOMI), 4. MR 0457337
- B. R. Schratzberger, The quality of approximation of Brun’s algorithm in three dimensions, Monatsh. Math. 134 (2001), no. 2, 143–157. MR 1878076, DOI 10.1007/s006050170004
- Fritz Schweiger, The metrical theory of Jacobi-Perron algorithm, Lecture Notes in Mathematics, Vol. 334, Springer-Verlag, Berlin-New York, 1973. MR 0345925
- Fritz Schweiger, Multidimensional continued fractions, Oxford Science Publications, Oxford University Press, Oxford, 2000. MR 2121855
- F. Schweiger, Invariant measure and exponent of convergence for Baldwin’s algorithm GCFP, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 210 (2001), 11–23 (2002). MR 2005886
- Fritz Schweiger, Ergodic and Diophantine properties of algorithms of Selmer type, Acta Arith. 114 (2004), no. 2, 99–111. MR 2068851, DOI 10.4064/aa114-2-1
- Ernst S. Selmer, Continued fractions in several dimensions, Nordisk Mat. Tidskr. 9 (1961), 37–43, 95 (Norwegian, with English summary). MR 130852
Additional Information
- Valérie Berthé
- Affiliation: Université de Paris, IRIF, CNRS, F–75006 Paris, France
- ORCID: 0000-0001-5561-7882
- Email: berthe@irif.fr
- Wolfgang Steiner
- Affiliation: Université de Paris, IRIF, CNRS, F–75006 Paris, France
- MR Author ID: 326598
- Email: steiner@irif.fr
- Jörg M. Thuswaldner
- Affiliation: Chair of Mathematics and Statistics, University of Leoben, A–8700 Leoben, Austria
- MR Author ID: 612976
- ORCID: 0000-0001-5308-762X
- Email: joerg.thuswaldner@unileoben.ac.at
- Received by editor(s): October 19, 2019
- Received by editor(s) in revised form: August 7, 2020
- Published electronically: November 16, 2020
- Additional Notes: This work was supported by the Agence Nationale de la Recherche through the project Codys (ANR-18-CE40-0007).
The third author was supported by the projects FWF P27050 and FWF P29910 granted by the Austrian Science Fund and by project FWF/RSF I3466 granted by the Austrian Science Fund and the Russian Science Foundation.
Part of this work has been done while the three authors were visiting the Erwin Schrödinger Institute in Vienna. - © Copyright 2020 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 883-905
- MSC (2020): Primary 11A55, 11J70, 11K50, 37D25
- DOI: https://doi.org/10.1090/mcom/3592
- MathSciNet review: 4194166