A compactum
X
⊂
C
X\subset \mathbb {C}
is unshielded if it coincides with the boundary of the unbounded component of
C
∖
X
\mathbb {C}\setminus X
. Call a compactum
X
X
finitely Suslinian if every collection of pairwise disjoint subcontinua of
X
X
whose diameters are bounded away from zero is finite. We show that any unshielded planar compactum
X
X
admits a topologically unique monotone map
m
X
:
X
→
X
F
S
m_X:X \to X_{FS}
onto a finitely Suslinian quotient such that any monotone map of
X
X
onto a finitely Suslinian quotient factors through
m
X
m_X
. We call the pair
(
X
F
S
,
m
X
)
(X_{FS},m_X)
(or, more loosely,
X
F
S
X_{FS}
) the finest finitely Suslinian model of
X
X
.
If
f
:
C
→
C
f:\mathbb {C}\to \mathbb {C}
is a branched covering map and
X
⊂
C
X \subset \mathbb {C}
is a fully invariant compactum, then the appropriate extension
M
X
M_X
of
m
X
m_X
monotonically semiconjugates
f
f
to a branched covering map
g
:
C
→
C
g:\mathbb {C}\to \mathbb {C}
which serves as a model for
f
f
. If
f
f
is a polynomial and
J
f
J_f
is its Julia set, we show that
m
X
m_X
(or
M
X
M_X
) can be defined on each component
Z
Z
of
J
f
J_f
individually as the finest monotone map of
Z
Z
onto a locally connected continuum.