Lavaurs algorithm for cubic symmetric polynomials

Ergodic Theory and Dynamical Systems. 2025. Vol. 45. No. 8. P. 2314–2340.

Blokh A., Oversteegen L., Selinger N., Timorin V., Vejandla S. C.

As discovered by W. Thurston, the action of a complex one-variable polynomial on its Julia set can be modeled by a geodesic lamination in the disk, provided that the Julia set is connected. It also turned out that the parameter space of such dynamical laminations of degree two gives a model for the bifurcation locus in the space of quadratic polynomials. This model is itself a geodesic lamination, the so called quadratic minor lamination of Thurston. In the same spirit, we consider the space of all cubic symmetric polynomials f λ (z) = z3 + λ2z in three articles. In the first one, we construct the cubic symmetric comajor lamination together with the corresponding quotient space of the unit circle. As is verified in the third paper, this yields a monotone model of the cubic symmetric
connectedness locus, that is, the space of all cubic symmetric polynomials with connected Julia sets. In the present paper, the second in the series, we develop an algorithm for generating the cubic symmetric comajor lamination analogous to the Lavaurs algorithm for constructing the quadratic minor lamination.

Keywords: Julia set, cubic polynomial, инвариантные ламинации, invariant laminations, множество Жюлиа, кубический многочлен

Publication based on the results of: Geometry of algebraic varieties, derived categories, motives, and metrics of special holonomy (2025)