Related papers: Almost every real quadratic polynomial has a poly-…
We present the first example of a poly-time computable Julia set with a recurrent critical point: we prove that the Julia set of the Feigenbaum map is computable in polynomial time.
In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable. We also show that a filled Julia set of a…
In this paper we prove that parabolic Julia sets of rational functions are locally computable in polynomial time.
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…
It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has…
We prove that the Julia set of a rational function $f$ is computable in polynomial time, assuming that the postcritical set of $f$ does not contain any critical points or parabolic periodic orbits.
We prove the existence of quadratic polynomials having a Julia set with positive Lebesgue measure in three cases: the presence of a Cremer fixed point, the presence of a Siegel disk, the presence of infinitely many (satellite)…
We show that if a polynomial filled Julia set has empty interior, then it is computable.
Since the 1980s, much progress has been done in completely determining which functions share a Julia set. The polynomial case was completely solved in 1995, and it was shown that the symmetries of the Julia set play a central role in…
The dynamics of all quadratic Newton maps of rational functions are completely described. The Julia set of such a map is found to be either a Jordan curve or totally disconnected. It is proved that no Newton map with degree at least three…
We find an abundance of Cremer Julia sets of an arbitrarily high computational complexity.
We extend Sullivan's complex a priori bounds to real quadratic polynomials with essentially bounded combinatorics. Combined with the previous results of the first author, this yields complex bounds for all real quadratics. Local…
We show that under the definition of computability which is natural from the point of view of applications, there exist non-computable quadratic Julia sets.
By a symmetry of the Julia set of a polynomial, also referred as polynomial Julia set, we mean an Euclidean isometry preserving the Julia set. Each such symmetry is in fact a rotation about the centroid of the polynomial. In this article, a…
The long-standing problem of existence of nowhere dense rational Julia set with positive area has been solved by an example in quadratic polynomials by Buff and Ch\'eritat. Since then many efforts have been devoted to finding out new…
We discuss computability of impressions of prime ends of compact sets. In particular, we construct quadratic Julia sets which possess explicitly described non-computable impressions.
We prove that almost every non-regular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps have excellent ergodic properties, as…
Let $f$ be a rational map with degree $d\geq 2$ whose Julia set is connected but not equal to the whole Riemann sphere. It is proved that there exists a rational map $g$ such that $g$ contains a buried Julia component on which the dynamics…
We prove that every wandering Julia component of cubic rational maps eventually has at most two complementary components.
We completely characterize the conformal radii of Siegel disks in the family $$P_\theta(z)=e^{2\pi i\theta}z+z^2,$$ corresponding to {\bf computable} parameters $\theta$. As a consequence, we constructively produce quadratic polynomials…