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We give an introduction to buried points in Julia sets and a list of questions about buried points, written to encourage aficionados of topology and dynamics to work on these questions.
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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 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 monotone map $psi:Jto J$ there exists a monotone map $h$ with $psi=hcirc ph$. Then we extend $ph$ onto the complex plane $C$ (keeping the same notation) and show that $ph$ monotonically semiconjugates $P|_{C}$ to a emph{topological polynomial $g:Cto C$}. If $P$ does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwis fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a criterion and a useful sufficient condition for the map $ph$ not to collapse $J$ into a point.
A. Sannami constructed an example of the differentiable Cantor set embedded in the real line whose difference set has a positive measure. In this paper, we generalize the definition of the difference sets for sets of the two dimensional Euclidean space as the sets of vectors between two sets, and estimate their measures. For the quadratic map Q_c(z)=z^2+c, we obtain that the measure of the difference set of its Julia set vanishes if |c|>3+sqrt{3}.
We find an abundance of Cremer Julia sets of an arbitrarily high computational complexity.
We construct the first examples of rational functions defined over a non-archimedean field with certain dynamical properties. In particular, we find such functions whose Julia sets, in the Berkovich projective line, are connected but not contained in a line segment. We also show how to compute the measure-theoretic and topological entropy of such maps. In particular, we show for some of our examples that the measure-theoretic entropy is strictly smaller than the topological entropy, thus answering a question of Favre and Rivera-Letelier.
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