Dynamics of solutions to quadratic forms on R3

Simons Semester Continued Fractions in Fractals, Ergodic theory and Dynamics Conference “Ergodic theory, fractal geometry and Diophantine approximation” Warsaw, May 18 – 22, 2026

Talk by: Alden Paige Dynamics of solutions to quadratic forms on R3

It is known that every positive primitive Pythagorean triple can be uniquely express in the form Ma1. . . Manv, where each Mai is one of three specific matrices, and v is either (3, 4, 5)T or (4, 3, 5)T. Motivated by a desire to
compute this code for any given triple, Romik presented an ergodic dynamical system on the positive quadrant of the unit circle, and conjugated this to the unit interval. Later, Cha (et al.) presented a method for com-
puting Berggren trees for some quadratic forms on R 3 that satisfy certain conditions. In this talk, I present a methodology for constructing 1-D dynamical systems from these Berggren trees following the outline of Romik’s paper, and give their absolutely continuous invariant measures by adapting Keane’s method of computing the Gauss measure for the continued fraction map. Time permitting, I will also show how the Farey map may be
exhibited as an example.

This lecture was partially supported by the Simons Foundation grant (award no. SFI-MPS-T-Institutes-00010825) and from State Treasury funds as part of a task commissioned by the Minister of Science and Higher Education under the project “Organization of the Simons Semesters at the Banach Center - New Energies in 2026-2028” (agreement no. MNiSW/2025/DAP/491).

Видео Dynamics of solutions to quadratic forms on R3 канала Banach Center