Speaker
Description
It is known that every positive primitive Pythagorean triple can be uniquely express in the form $M_{a_1} \ldots M_{a_n} v$, where each $M_{a_i}$ 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 computing Berggren trees for some quadratic forms on $\mathbb{R}^3$ 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.