Finitely Suslinian models for planar compacta with applications to Julia sets

Abstract:

A compactum $X\subset \mathbb {C}$ is unshielded if it coincides with the boundary of the unbounded component of $\mathbb {C}\setminus X$. 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 $m_X:X \to X_{FS}$ onto a finitely Suslinian quotient such that any monotone map of $X$ onto a finitely Suslinian quotient factors through $m_X$. We call the pair $(X_{FS},m_X)$ (or, more loosely, $X_{FS}$) the finest finitely Suslinian model of $X$.

If $f:\mathbb {C}\to \mathbb {C}$ is a branched covering map and $X \subset \mathbb {C}$ is a fully invariant compactum, then the appropriate extension $M_X$ of $m_X$ monotonically semiconjugates $f$ to a branched covering map $g:\mathbb {C}\to \mathbb {C}$ which serves as a model for $f$. If $f$ is a polynomial and $J_f$ is its Julia set, we show that $m_X$ (or $M_X$) can be defined on each component $Z$ of $J_f$ individually as the finest monotone map of $Z$ onto a locally connected continuum.