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In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impression.
We show that if $P$ is a quadratic polynomial with a fixed Cremer point and Julia set $J$, then for any monotone map $ph:Jto A$ from $J$ onto a locally connected continuum $A$, $A$ is a single point.
Let $P$ be a polynomial of degree $d$ with a Cremer point $p$ and no repelling or parabolic periodic bi-accessible points. We show that there are two types of such Julia sets $J_P$. The emph{red dwarf} $J_P$ are nowhere connected im kleinen and such
We find an abundance of Cremer Julia sets of an arbitrarily high computational complexity.
Let $P$ be a polynomial with a connected Julia set $J$. We use continuum theory to show that it admits a emph{finest monotone map $ph$ onto a locally connected continuum $J_{sim_P}$}, i.e. a monotone map $ph:Jto J_{sim_P}$ such that for any other mon
We show that there exist real parameters $c$ for which the Julia set $J_c$ of the quadratic map $z^2+c$ has arbitrarily high computational complexity. More precisely, we show that for any given complexity threshold $T(n)$, there exist a real paramete