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Studying the albedos of the planets and moons of the Solar System dates back at least a century. Of particular interest is the relationship between the albedo measured at superior conjunction, known as the ``geometric albedo", and the…
Solar prominences and filaments (prominences projected against the solar disk) exhibit a large variety of fine structures which are well observed down to the resolution limit of ground-based telescopes. We describe the morphological aspects…
We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as {\em spinning}. We show that under fairly general hypotheses, the limit of spinning either exists and is unique, or else converges…
This article studies algebraic elements of the Cremona group. In particular, we show that the set of all these elements is a countable union of closed subsets but it is not closed.
We survey the definition of the radial Julia set of a meromorphic function (in fact, more generally, any "Ahlfors islands map"), and give a simple proof that the Hausdorff dimension of the reduced Julia set always coincides with the…
It is known that Laplacian operators on many fractals have gaps in their spectra. This fact precludes the possibility that a Weyl-type ratio can have a limit and is also a key ingredient in proving that the Fourier series on such fractals…
Coronal plumes are narrow, collimated structures that are primarily viewed above the solar poles and in coronal holes in the extreme ultraviolet, but also in sunspots. Open questions remain about plume formation, including the role of…
We show, in an elementary way, that the Julia set of one-complex-variable entire functions is nonempty and perfect.
We prove that the Julia set of a rational map of the Riemann sphere satisfying the Collet-Eckmann condition and having no parabolic periodic point is mean porous, if it is not the whole sphere. It follows that the Minkowski dimension of the…
We give a method for constructing many examples of automorphisms with positive entropy on rational complex surfaces. The general idea is to begin with a quadratic Cremona transformation that fixes a reduced cubic curve and then use the…
Given a critically periodic quadratic map with no secondary renormalizations, we introduce the notion of $Q$-recurrent quadratic polynomials. We show that the pieces of the principal nest of a $Q$-recurrent map $f_c$ converge in shape to…
It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes, this result comes down to the fact that such a Julia set is…
We prove that the Julia set $J(f)$ of at most finitely renormalizable unicritical polynomial $f:z\mapsto z^d+c$ with all periodic points repelling is locally connected. (For $d=2$ it was proved by Yoccoz around 1990.) It follows from a…
A model for the Mandelbrot set is due to Thurston and is stated in the language of geodesic laminations. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating…
We describe the statistical properties of the dynamics of the quadratic polynomials P_a(z):=e^{2\pi a i} z+z^2 on the complex plane, with a of high return times. In particular, we show that these maps are uniquely ergodic on their measure…
We study how the orbits of the singularities of the inverse of a meromorphic function prescribe the dynamics on its Julia set, at least up to a set of (Lebesgue) measure zero. We concentrate on a family of entire transcendental functions…
We study complex one-dimensional parameter slices in a three-parameter family of rational maps with two free critical points, obtained by imposing the existence of periodic orbits with prescribed multipliers. Using explicit…
Any planar shape $P\subset \mathbb{C}$ can be embedded isometrically as part of the boundary surface $S$ of a convex subset of $\mathbb{R}^3$ such that $\partial P$ supports the positive curvature of $S$. The complement $Q = S \setminus P$…
The study of Julia sets gives a new and natural way to look at fractals. When mathematicians investigated the special class of Misiurewicz's rational maps, they found out that there is a Julia set which is homeomorphic to a well known…
We study the closure of the cubic Principal Hyperbolic Domain and its intersection $\mathcal{P}_\lambda$ with the slice $\mathcal{F}_\lambda$ of the space of all cubic polynomials with fixed point $0$ defined by the multiplier $\lambda$ at…