Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The dynamics of Pythagorean Triples

Author: Dan Romik
Journal: Trans. Amer. Math. Soc. 360 (2008), 6045-6064
MSC (2000): Primary 37A45
Published electronically: April 22, 2008
MathSciNet review: 2425702
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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. Jon Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol. 50, American Mathematical Society, Providence, RI, 1997. MR 1450400
  • 2. Jon Aaronson and Manfred Denker, The Poincaré series of 𝐂\sbs𝐙, Ergodic Theory Dynam. Systems 19 (1999), no. 1, 1–20. MR 1676950, 10.1017/S0143385799126592
  • 3. Roy L. Adler, Geodesic flows, interval maps, and symbolic dynamics, Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), Oxford Sci. Publ., Oxford Univ. Press, New York, 1991, pp. 93–123. MR 1130174
  • 4. Roy L. Adler and Leopold Flatto, Cross section maps for geodesic flows. I. The modular surface, Ergodic theory and dynamical systems, II (College Park, Md., 1979/1980), Progr. Math., vol. 21, Birkhäuser, Boston, Mass., 1982, pp. 103–161. MR 670077
  • 5. Roy L. Adler and Leopold Flatto, The backward continued fraction map and geodesic flow, Ergodic Theory Dynam. Systems 4 (1984), no. 4, 487–492. MR 779707, 10.1017/S0143385700002583
  • 6. Roy L. Adler and Leopold Flatto, Cross section map for the geodesic flow on the modular surface, Conference in modern analysis and probability (New Haven, Conn., 1982), Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 9–24. MR 737384, 10.1090/conm/026/737384
  • 7. Roger C. Alperin, The modular tree of Pythagoras, Amer. Math. Monthly 112 (2005), no. 9, 807–816. MR 2179860, 10.2307/30037602
  • 8. Viviane Baladi and Brigitte Vallée, Euclidean algorithms are Gaussian, J. Number Theory 110 (2005), no. 2, 331–386. MR 2122613, 10.1016/j.jnt.2004.08.008
  • 9. F. J. M. Barning, On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices, Math. Centrum Amsterdam Afd. Zuivere Wisk. 1963 (1963), no. ZW-011, 37 (Dutch). MR 0190077
  • 10. Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0192027
  • 11. Johannes Gollnick, Harald Scheid, and Joachim Zöllner, Rekursive Erzeugung der primitiven pythagoreischen Tripel, Math. Semesterber. 39 (1992), no. 1, 85–88 (German). MR 1161588, 10.1007/BF03186460
  • 12. A. Hall,
    Genealogy of Pythagorean triads.
    Math. Gazette 54:390 (1970), 377-379.
  • 13. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125
  • 14. Fritz Herzog and B. M. Stewart, Patterns of visible and nonvisible lattice points, Amer. Math. Monthly 78 (1971), 487–496. MR 0284403
  • 15. Jens Jaeger, Pythagorean number sets, Nordisk Mat. Tidskr. 24 (1976), no. 2, 56–60, 75 (Danish, with English summary). MR 0457332
  • 16. A. R. Kanga, The family tree of Pythagorean triples, Bull. Inst. Math. Appl. 26 (1990), no. 1-2, 15–17. MR 1040886
  • 17. Donald E. Knuth, The art of computer programming. Vol. 2, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms; Addison-Wesley Series in Computer Science and Information Processing. MR 633878
  • 18. Erik Kristensen, Pythagorean number sets and orthonormal matrices, Nordisk Mat. Tidskr. 24 (1976), no. 3-4, 111–122, 135 (1977) (Danish, with English summary). MR 0491471
  • 19. A. Lönnemo,
    The trinary tree underlying primitive pythagorean triples.
    In: Cut the Knot, Interactive Mathematics Miscellany and Puzzles. Alex Bogomolny (Ed.), http://www.
  • 20. Darryl McCullough, Height and excess of Pythagorean triples, Math. Mag. 78 (2005), no. 1, 26–44. MR 2126355, 10.2307/3219271
  • 21. Paul Préau, Un graphe ternaire associé à l’équation 𝑋²+𝑌²=𝑍², C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 7, 665–668 (French, with English and French summaries). MR 1300066
  • 22. A. Redmon,
    Pythagorean triples,
    Master's Thesis, Eastern Washington University, 1998.
  • 23. Fritz Schweiger, Ergodic theory of fibred systems and metric number theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. MR 1419320
  • 24. Caroline Series, The modular surface and continued fractions, J. London Math. Soc. (2) 31 (1985), no. 1, 69–80. MR 810563, 10.1112/jlms/s2-31.1.69
  • 25. Caroline Series, Geometrical methods of symbolic coding, Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), Oxford Sci. Publ., Oxford Univ. Press, New York, 1991, pp. 125–151. MR 1130175
  • 26. N. J. A. Sloane, editor (2003),
    The On-Line Encyclopedia of Integer Sequences.
    http://www. njas/sequences/.
  • 27. Brigitte Vallée, Dynamical analysis of a class of Euclidean algorithms, Theoret. Comput. Sci. 297 (2003), no. 1-3, 447–486. Latin American theoretical informatics (Punta del Este, 2000). MR 1981160, 10.1016/S0304-3975(02)00652-7
  • 28. A. Wayne,
    A genealogy of $ 120^{o}$ and $ 60^{o}$ triangles. Math. Mag. 55 (1982), 157-162.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37A45

Retrieve articles in all journals with MSC (2000): 37A45

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

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.