A compactum X ⊂ C is unshielded if it coincides with the boundary of the unbounded component of C\X. Call a compactum X finitely Suslinian if every collection of pairwise disjoint subcontinua of X whose diameters are bounded away from zero is finite. We show that any unshielded planar compactum X admits a topologically unique monotone map mX: X → XFS onto a finitely Suslinian quotient such that any monotone map of X onto a finitely Suslinian quotient factors through mX. We call the pair (XFS, mX) (or, more loosely, XFS) the finest finitely Suslinian model of X. If f: ℂ → ℂ is a branched covering map and X ⊂ ℂ is a fully invariant compactum, then the appropriate extension MX of mX monotonically semiconjugates f to a branched covering map g: ℂ → ℂ which serves as a model for f. If f is a polynomial and Jf is its Julia set, we show that mX (or MX) can be defined on each component Z of Jf individually as the finest monotone map of Z onto a locally connected continuum. © 2012 American Mathematical Society.
