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The dynamics of Pythagorean Triples


Author: Dan Romik
Journal: Trans. Amer. Math. Soc. 360 (2008), 6045-6064
MSC (2000): Primary 37A45
DOI: https://doi.org/10.1090/S0002-9947-08-04467-X
Published electronically: April 22, 2008
MathSciNet review: 2425702
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Abstract: We construct a piecewise onto 3-to-1 dynamical system on the positive quadrant of the unit circle, such that for rational points (which correspond to normalized Primitive Pythagorean Triples), the associated ternary expansion is finite and is equal to the address of the PPT on Barning's (1963) ternary tree of PPTs, while irrational points have infinite expansions. The dynamical system is conjugate to a modified Euclidean algorithm. The invariant measure is identified, and the system is shown to be conservative and ergodic. We also show, based on a result of Aaronson and Denker (1999), that the dynamical system can be obtained as a factor map of a cross section of the geodesic flow on a quotient space of the hyperbolic plane by the group $ \Gamma(2)$, a free subgroup of the modular group with two generators.


References [Enhancements On Off] (What's this?)

  • 1. J. Aaronson,
    An Introduction to Infinite Ergodic Theory.
    Mathematical Surveys and Monographs 50,
    Amer. Math. Soc, Providence, RI, 1997. MR 1450400 (99d:28025)
  • 2. J. Aaronson, M. Denker,
    The Poincaré series of $ \mathbb{C}\setminus\mathbb{Z}$.
    Ergodic Theory Dynam. Systems 19 (1999), 1-20. MR 1676950 (2001b:37042)
  • 3. R. L. Adler,
    Geodesic flows, interval maps, and symbolic dynamics.
    In: Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, 93-123.
    Oxford University Press, Oxford, 1991. MR 1130174
  • 4. R. L. Adler, L. Flatto,
    Cross section maps for geodesic flows. I. The modular surface.
    In: Ergodic theory and dynamical systems, vol. 2 (College Park, Maryland, 1979-1980), 103-161.
    Progr. Math., 21, Birkhäuser, Boston, Mass., 1982. MR 670077 (84h:58113)
  • 5. R. L. Adler, L. Flatto,
    The backward continued fraction map and geodesic flow.
    Ergodic Theory Dynam. Systems 4 (1984), 487-492. MR 779707 (86h:58116)
  • 6. R. L. Adler, L. Flatto,
    Cross section map for the geodesic flow on the modular surface.
    In: Conference in modern analysis and probability (New Haven, Conn., 1982), 9-24.
    Contemp. Math., 26, Amer. Math. Soc., Providence, RI, 1984. MR 737384 (85j:58128)
  • 7. R. Alperin,
    The Modular tree of Pythagoras.
    Amer. Math. Monthly 112 (2005), 807-816. MR 2179860 (2006h:11029)
  • 8. V. Baladi, B. Valleé,
    Euclidean algorithms are Gaussian. J. Number Theory 110 (2005), 331-386. MR 2122613 (2006e:11192)
  • 9. F. J. M. Barning,
    On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices. (Dutch)
    Math. Centrum Amsterdam Afd. Zuivere Wisk., ZW-011 (1963). MR 0190077 (32:7491)
  • 10. P. Billingsley,
    Ergodic Theory and Information.
    John Wiley & Sons, New York-London-Sydney 1965. MR 0192027 (33:254)
  • 11. J. Gollnick, H. Scheid, J. Zöllner,
    Rekursive Erzeugung der primitiven pythagoreischen Tripel. (German)
    Math. Semesterber. 39 (1992), 85-88. MR 1161588 (93c:11012)
  • 12. A. Hall,
    Genealogy of Pythagorean triads.
    Math. Gazette 54:390 (1970), 377-379.
  • 13. G. H. Hardy, E. M. Wright,
    An Introduction to the Theory of Numbers, 5th ed.
    Oxford University Press, Oxford, 1985. MR 0067125 (16:673c)
  • 14. F. Herzog, B. M. Stewart, Patterns of visible and nonvisible lattice points. Amer. Math. Monthly 78 (1971), 487-496. MR 0284403 (44:1630)
  • 15. J. Jaeger,
    Pythagorean number sets. (Danish.)
    Nordisk Mat. Tidskr. 24 (1976), 56-60, 75. MR 0457332 (56:15540)
  • 16. A. R. Kanga,
    The family tree of Pythagorean triples.
    Bull. Inst. Math. Appl. 26 (1990), 15-17. MR 1040886 (91b:11026)
  • 17. D. E. Knuth,
    The Art of Computer Programming, vol. 2: Seminumerical Algorithms, 3rd. ed.
    Addison-Wesley, 1998. MR 633878 (83i:68003)
  • 18. E. Kristensen,
    Pythagorean number sets and orthonormal matrices. (Danish)
    Nordisk Mat. Tidskr. 24 (1976), 111-122, 135. MR 0491471 (58:10717)
  • 19. A. Lönnemo,
    The trinary tree underlying primitive pythagorean triples.
    In: Cut the Knot, Interactive Mathematics Miscellany and Puzzles. Alex Bogomolny (Ed.), http://www. cut-the-knot.org/pythagoras/PT_matrix.shtml.
  • 20. D. McCullough,
    Height and excess of Pythagorean triples.
    Math. Mag. 78 (2005), 26-44. MR 2126355
  • 21. P. Préau,
    Un graphe ternaire associé à l'équation $ X\sp 2+Y\sp 2=Z\sp 2$.
    C. R. Acad. Sci. Paris Ser. I Math. 319 (1994), 665-668. MR 1300066 (95h:11023)
  • 22. A. Redmon,
    Pythagorean triples,
    Master's Thesis, Eastern Washington University, 1998.
  • 23. F. Schweiger,
    Ergodic Theory of Fibred Systems and Metric Number Theory.
    Clarendon Press, Oxford, 1995. MR 1419320 (97h:11083)
  • 24. C. Series,
    The modular surface and continued fractions.
    J. London Math. Soc. (2) 31 (1985), 69-80. MR 810563 (87c:58094)
  • 25. C. Series,
    Geometrical methods of symbolic coding.
    In: Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, 125-151.
    Oxford University Press, Oxford, 1991. MR 1130175
  • 26. N. J. A. Sloane, editor (2003),
    The On-Line Encyclopedia of Integer Sequences.
    http://www. research.att.com/ njas/sequences/.
  • 27. B. Vallée,
    Dynamical analysis of a class of Euclidean algorithms.
    Theor. Comp. Sci. 297 (2003), 447-486. MR 1981160 (2004e:11144)
  • 28. A. Wayne,
    A genealogy of $ 120^{o}$ and $ 60^{o}$ triangles. Math. Mag. 55 (1982), 157-162.

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Additional Information

Dan Romik
Affiliation: Department of Statistics, University of California at Berkeley, 367 Evans Hall, Berkeley, California 94720-3860
Address at time of publication: Einstein Institute of Mathematics, The Hebrew University, Givat-Ram, Jerusalem 91904, Israel
Email: romik@math.huji.ac.il

DOI: https://doi.org/10.1090/S0002-9947-08-04467-X
Received by editor(s): May 26, 2006
Received by editor(s) in revised form: November 13, 2006
Published electronically: April 22, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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