The dynamics of Pythagorean triples

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 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 cross-section of the geodesic flow on a quotient space of the hyperbolic plane by the group $Γ(2)$, a free subgroup of the modular group with two generators.