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Transactions of the American Mathematical Society
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
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-08-04467-X
PII: S 0002-9947(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.