Dan Romik (Weizmann I.)

will speak on

The dynamical system of Pythagorean triples.

Abstract:

In the 1960's, Barning constructed three 3x3 matrices A,B,C, such that every primitive Pythagorean triple (a,b,c), a^2+b^2=c^2, with a odd and b even, can be represented in a unique way as a product of these matrices in some order times the fundamental triple (3,4,5) considered as a column vector. By mapping the triple (a,b,c) to the unit vector (a/c,b/c), this leads to a natural dynamical system on the positive quadrant of the unit circle, a variant of the continued fraction family of mappings, whose properties we will study. In particular, we identify the absolutely continuous invariant measure, show that the resulting measure preserving system is conservative and ergodic, and show that it can be obtained as a factor of a section of the geodesic flow on the quotient of the hyperbolic plane by the congruence subgroup Gamma(2), a freely generated subgroup of SL(2,Z) with two generators.

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Last update on 20/10/02.