Michael Yampolsky

Non-computable Julia sets

Polynomial Julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system. Apart from the beautiful mathematics, one of the reasons for their popularity is the beauty of the computer-generated images of such sets. The algorithms used to draw these pictures vary; the most naive work by iterating the center of a pixel to determine if it lies in the Julia set. Milnors distance-estimator algorithm [Mil] uses classical complex analysis to give a one-pixel estimate of the Julia set. This algorithm and its modifications work quite well for many examples, but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution. Moreover, there are examples, even in the family of quadratic polynomials, when no satisfactory pictures of the Julia set exist. In this paper we study computability properties of Julia sets of quadratic polynomials. Under the definition we use, a set is computable, if, roughly speaking, its image can be generated by a computer with an arbitrary precision. Under this notion of computability we show:

Mating Siegel quadratic polynomials

1.1. Mating: Definitions and some history. Mating quadratic polynomials is a topological construction suggested by Douady and Hubbard [Do2] to partially parametrize quadratic rational maps of the Riemann sphere by pairs of quadratic polynomials. Some results on matings of higher degree maps exist, but we will not discuss them in this paper. While there exist several, presumably equivalent, ways of describing the construction of mating, the following approach is perhaps the most standard. Consider two monic quadratic polynomials fi and f2 whose filled Julia sets K(fi) are locally-connected. For each fi, let (Di denote the conformal isomorphism between the basin of infinity C -,. K(f,) and C -. ED, with i(00) = 00 and V(oo) = 1. These Bottcher maps conijugate the polynomials to the squaring map:

The Attractor of Renormalization¶and Rigidity of Towers of Critical Circle Maps

Abstract: We demonstrate the existence of a global attractor ? with a Cantor set structure for the renormalization of critical circle mappings. The set ? is invariant under a generalized renormalization transformation, whose action on ? is conjugate to the two-sided shift with a countable alphabet.

Metabolic scaling law for fetus and placenta

The human fetal birth weight does not scale linearly with the weight of the placenta, but exhibits an allometric scaling consistent with Kleibers Law for the basal metabolic rate. We discuss the possible causes of such scaling, and its clinical consequences. In particular, we show that the value of the scaling exponent is an indicator of a normal fetoplacental development.

Filled Julia Sets with Empty Interior Are Computable

We show that if a polynomial filled Julia set has empty interior, then it is computable.

Geography of the cubic connectedness locus: Intertwining surgery

Abstract We exhibit products of Mandelbrot sets in the two-dimensional complex parameter space of cubic polynomials. Cubic polynomials in such a product may be renormalized to produce a pair of quadratic maps. The inverse construction intertwining two quadratics is realized by means of quasiconformal surgery. The associated asymptotic geography of the cubic connectedness locus is discussed in the Appendix.

Cylinder renormalization of siegel disks

We study one of the central open questions in one-dimensional renormalization theory—the conjectural universality of goldenmean Siegel disks. We present an approach to the problem based on cylinder renormalization proposed by the second author. Numerical implementation of this approach relies on the constructive measurable Riemann mapping theorem proved by the first author. Our numerical study yields convincing evidence to support the hyperbolicity conjecture in this setting.

On Computational Complexity of Siegel Julia Sets

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 polynomial complexity. In this paper we demonstrate the existence of computable quadratic Julia sets whose computational complexity is arbitrarily high.

Constructing non-computable Julia sets

While most polynomial Julia sets are computable, it has been recently shown [12] that there exist non-computable Julia sets. The proof was non-constructive, and indeed there were doubts as to whether specific examples of parameters with non-computable Julia sets could be constructed. It was also unknown whether the non-computability proof can be extended to the filled Julia sets. In this paper we give an answer to both of these questions, which were the main open problems concerning the computability of polynomial Julia sets. We show how to construct a specific polynomial with a non-computable Julia set. In fact, in the case of Julia sets of quadratic polynomials we give a precise characterization of Juliasets with computable parameters. Moreover, assuming a widely believed conjecture in Complex Dynamics, we give a poly-time algorithm forcomputing a number c such that the Julia set Jz2+c z is non-computable. In contrast with these results, we show that the filled Julia set of a polynomial is always computable.

Holomorphic dynamics and renormalization : a volume in honour of John Milnor's 75th birthday

Schwarzian derivatives and cylinder maps by A. Bonifant and J. Milnor Holomorphic dynamics: Symbolic dynamics and self-similar groups by V. Nekrashevych Are there critical points on the boundaries of mother hedgehogs? by D. K. Childers Finiteness for degenerate polynomials by L. DeMarco Cantor webs in the parameter and dynamical planes of rational maps by R. L. Devaney Simple proofs of uniformization theorems by A. A. Glutsyuk The Yoccoz combinatorial analytic invariant by C. L. Petersen and P. Roesch Bifurcation loci of exponential maps and quadratic polynomials: Local connectivity, triviality of fibers, and density of hyperbolicity by L. Rempe and D. Schleicher Rational and transcendental Newton maps by J. Ruckert Newtons method as a dynamical system: Efficient root finding of polynomials and the Riemann