Hostname: page-component-cb9f654ff-plnhv Total loading time: 0.001 Render date: 2025-08-16T22:03:16.710Z Has data issue: false hasContentIssue false
  • English
  • Français

Ergodicity and algebraicity of the fast and slow triangle maps

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Diophantine approximation, transcendental number theory Algebraic number theory: global fields Measure-theoretic ergodic theory

Published online by Cambridge University Press:  08 August 2025

THOMAS GARRITY  and
JACOB LEHMANN DUKE
THOMAS GARRITY*
Affiliation:
Department of Mathematics, https://ror.org/04avkmd49Williams College, Williamstown 01267, MA, USA
JACOB LEHMANN DUKE
Affiliation:
Department of Mathematics, https://ror.org/049s0rh22 Dartmouth College , Hannover 03755, NH, USA (e-mail: jacob.lehmann.duke.gr@dartmouth.edu)
*
e-mail: tgarrity@williams.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Our goal is to show that both the fast and slow versions of the triangle map (a type of multi-dimensional continued fraction algorithm) in dimension n are ergodic, resolving a conjecture of Messaoudi, Noguiera, and Schweiger [Ergodic properties of triangle partitions. Monatsh. Math. 157 (2009), 283–299]. This particular type of higher dimensional multi-dimensional continued fraction algorithm has recently been linked to the study of partition numbers, with the result that the underlying dynamics has combinatorial implications.

Keywords

multi-dimensional continued fractiontriangle mapergodicity

MSC classification

Primary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Secondary: 11J70: Continued fractions and generalizations 11R04: Algebraic numbers; rings of algebraic integers 28D99: None of the above, but in this section

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 35
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Andrews, G.. The Theory of Integer Partitions. Cambridge University Press, Cambridge, 1998 (1st edn, 1976).Google Scholar
Assaf, S., Chen, L.-C., Cheslack-Postava, T., Cooper, B., Diesl, A., Garrity, T., Lepinski, M. and Schuyler, A.. A dual approach to triangle sequences: a multidimensional continued fraction algorithm. Integers 5 (2005), A08.Google Scholar
Baalbaki, W., Bonanno, C., Del Vigna, A., Garrity, T. and Isola, S.. On integer partitions and continued fraction type algorithms. Ramanujan J. 63(3) (2024), 873–915.CrossRefGoogle Scholar
Baalbaki, W. and Garrity, T.. Generating new partition identities via a generalized continued fraction algorithm. Electron. J. Combin. 31(1) (2024), Paper no. 1.20.10.37236/12178CrossRefGoogle Scholar
Baladi, V.. Positive Transfer Operators and Decay of Correlation (Advanced Series in Nonlinear Dynamics, 16). World Scientific, Singapore, 2000.10.1142/3657CrossRefGoogle Scholar
Berthé, V., Steiner, W. and Thuswaldner, J.. On the second Lyapunov exponent of some multidimensional continued fraction algorithms. Math. Comp. 90(328) (2021), 883–905.10.1090/mcom/3592CrossRefGoogle Scholar
Billingsley, P.. Ergodic Theory and Information. John Wiley & Sons, Inc., New York, 1965.Google Scholar
Bonanno, C. and Del Vigna, A.. Representation and coding of rational pairs on a Triangular tree and Diophantine approximation in ${\mathbb{R}}^2$ . Acta Arith. 200(4) (2021), 389–427.CrossRefGoogle Scholar
Bonanno, C., Del Vigna, A. and Munday, S.. A slow triangle map with a segment of indifferent fixed points and a complete tree of rational pairs. Monatsh. Math. 194(1) (2021), 1–40.Google Scholar
Dajani, K. and Kraaikamp, C.. Ergodic Theory of Numbers. The Mathematical Association of America, Washington, DC, 2002.CrossRefGoogle Scholar
Fougeron, C. and Skripchenko, A.. Simplicity of spectra for certain multidimensional continued fraction algorithms. Monatsh. Math. 194(4) (2021), 767–787.CrossRefGoogle Scholar
Fox, J. and Garrity, T.. Tree structure for integer partitions and the natural extension for the triangle map (a multi-dimensional continued fraction algorithm), in preparation.Google Scholar
Garrity, T.. On periodic sequences for algebraic numbers. J. Number Theory 88 (2001), 86–103.10.1006/jnth.2000.2608CrossRefGoogle Scholar
Garrity, T.. On Gauss–Kuzmin statistics and the transfer operator for a multidimensional continued fraction algorithm: the Triangle map. Publ. Math. Debrecen 101(3–4) (2022), 263–280.Google Scholar
Hensley, D.. Continued Fractions. World Scientific, Singapore, 2006.10.1142/5931CrossRefGoogle Scholar
Iosifescu, M. and Kraaikamp, C.. Metrical Theory of Continued Fractions (Mathematics and Its Applications, 547). Kluwer Academic Publishers, Dordrecht, 2002.10.1007/978-94-015-9940-5CrossRefGoogle Scholar
Ito, H.. Self-duality of multidimensional continued fractions. Preprint, 2022, arXiv:2203.07887.Google Scholar
Karpenkov, O.. Geometry of Continued Fractions (Algorithms and Computation in Mathematics, 26), 2nd edn. Springer, Berlin, 2022.CrossRefGoogle Scholar
Karpenkov, O.. On Hermite’s problem, Jacobi–Perron type algorithms, and Dirichlet groups. Acta Arith. 203(1) (2022), 27–48.CrossRefGoogle Scholar
Karpenkov, O.. On a periodic Jacobi–Perron type algorithm. Monatsh. Math. 205(3) (2024), 531–601.10.1007/s00605-024-02006-5CrossRefGoogle Scholar
Kesseböhmer, M., Munday, S. and Stratmann, B. O.. Infinite Ergodic Theory of Numbers. Walter de Gruyter GmbH, Berlin, 2016.Google Scholar
Messaoudi, A., Nogueira, A. and Schweiger, F.. Ergodic properties of triangle partitions. Monatsh. Math. 157 (2009), 283–299.CrossRefGoogle Scholar
Rockett, A. and Szusz, P.. Continued Fractions. World Scientific, Singapore, 1992.CrossRefGoogle Scholar
Schweiger, F.. Multidimensional Continued Fractions. Oxford University Press, Oxford, 2000.CrossRefGoogle Scholar
Schweiger, F.. Invariant simplices for subtractive algorithms. J. Number Theory 133 (2013), 2182–2185.10.1016/j.jnt.2012.12.005CrossRefGoogle Scholar
Silva, C. F.. Invitation to Ergodic Theory (Student Mathematical Library, 42). American Mathematical Society, Providence, RI, 2008.Google Scholar
Sloane, N. J. A.. Online Encyclopedia of Integer Sequences. https://oeis.org/A124774.Google Scholar