Dynamics of integer zeroes of homogeneous quadratic equations over $\mathbb{R}^3$

Romik has presented a construction of a 1-dimensional dynamical system on the unit interval by developing an algorithm that returns the unique sequence of matrices associated with a positive primitive Pythagorean triple (in the sense of Barning), and projecting the map involved in this algorithm onto an appropriate 1-dimensional space via stereographic projection. Romik additionally computes the infinite, absolutely continuous invariant measure, and shows that the system is conservative and ergodic. Later, Cha et al. provided a method of calculating "Berggren trees", which are generalisations of the tree of positive primitive Pythagorean triples one may construct via Barning's theorem, except for different homogeneous quadratic equations in 3 variables. We present here a method of computing 1-dimensional dynamical systems induced from these Berggren trees following Romik's outline, and determine their absolutely continuous invariant measures by adapting the method of Keane.