Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe solar Julia sets of basic quadratic Cremer polynomials
348
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 that the intersection of all impressions of external angles is a continuum containing $p$ and the orbits of all critical images. The emph{solar} $J_P$ are such that every angle with dense orbit has a degenerate impression disjoint from other impressions and $J_P$ is connected im kleinen at its landing point. We study bi-accessible points and locally connected models of $J_P$ and show that such sets $J_P$ appear through polynomial-like maps for generic polynomials with Cremer points.
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 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.
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 parameter $c$ such that the computational complexity of computing $J_c$ with $n$ bits of precision is higher than $T(n)$. This is the first known class of real parameters with a non poly-time computable Julia set.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
