In his 84 preprint W. Thurston proved that quadratic laminations do not admit so-called wandering triangles and asked a deep
question concerning their existence for laminations of higher
degrees. Recently it has been discovered by L. Oversteegen and the
author that some closed laminations of the unit circle invariant
under $z\mapsto z^d, d>2$ admit wandering triangles. This makes
the problem of describing the criteria for the existence of
wandering triangles important because solving this problem would
help understand the combinatorial structure of the family of all
polynomials of the appropriate degree.
In this paper for a closed lamination on the unit circle invariant
under $z\mapsto z^3$ (cubic lamination) we prove that if it has a
wandering triangle then there must be two distinct recurrent
critical points in the corresponding quotient space ("topological
Julia set") $J$ with the same limit set coinciding with the limit
set of any wandering vertex (wandering vertices in $J$ correspond
to wandering gaps in the lamination).