  A linear differential operator $T=Q(z)\frac{d}{dz}+P(z)$ with polynomial
coefficients defines a continuous family of Hutchinson operators when acting on
the space of positive powers of linear forms. In this context, $T$ has a unique
minimal Hutchinson-invariant set $M_{CH}^{T}$ in the complex plane. Using a
geometric interpretation of its boundary in terms of envelops of certain
families of rays, we subdivide this boundary into local and global arcs (the
former being portions of integral curves of the rational vector field
$\frac{Q(z)}{P(z)}\partial_{z}$), and singular points of different types which
we classify below. The latter decomposition of the boundary of $M_{CH}^{T}$ is
largely determined by its intersection with the plane algebraic curve formed by
the inflection points of trajectories of the field
$\frac{Q(z)}{P(z)}\partial_{z}$. We provide an upper bound for the number of
local arcs in terms of degrees of $P$ and $Q$. As an application of our
classification, we obtain a number of global geometric properties of minimal
Hutchinson-invariant sets.
