The dynamics of Pythagorean triples
Abstract
We construct a piecewise onto 3to1 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 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, that the dynamical system can be obtained as a factor map of a crosssection 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2004
 arXiv:
 arXiv:math/0406512
 Bibcode:
 2004math......6512R
 Keywords:

 Dynamical Systems;
 Number Theory
 EPrint:
 25 pages, 3 figures