Editor’s Note: This is the first installment of a two-part post by Amitai Etzioni examining the nation’s anti-obesity policies through the lens of a responsive communitarian philosophy. Today, Etzioni lays out a responsive communitarian framework and uses it to diagnose the problems with our current methods of fighting obesity. Tomorrow , Etzioni describes how these current policies should be r...

We prove that the Julia set J(f) of at most finitely renormalizable unicritical polynomial f : z 7→ z + c with all periodic points repelling is locally connected. (For d = 2 it was proved by Yoccoz around 1990.) It follows from a priori bounds in a modified Principle Nest of puzzle pieces. The proof of a priori bounds makes use of new analytic tools developed in [KL] that give control of moduli...

We consider the class of meromorphic functions whose set of singular values is bounded and for which the ω -limit set of the post singular set is a compact repeller. We show that if two simple growth conditions are satisfied, then the function is ergodic on its Julia set.

A complex map can give rise to two kinds of fractal sets: the Julia sets and the parameters sets (or the connectivity loci) which represent different connectivity properties of the corresponding Julia sets. In the significative results of (Int. J. Bifurc. Chaos, 2009, 19:2123–2129) and (Nonlinear. Dyn. 2013, 73:1155–1163), the authors presented the two kinds of fractal sets of a class of altern...

We show that, if the Julia set of a transcendental entire function is locally connected, then it takes the form of a spider’s web in the sense defined by Rippon and Stallard. In the opposite direction, we prove that a spider’s web Julia set is always locally connected at a dense subset of buried points. We also show that the set of buried points (the residual Julia set) can be a spider’s web.

where GK is the pluricomplex Green’s function of K with pole at infinity, d = ∂ + ∂̄, and d = (i/2π)(∂̄ − ∂). The support of μK is contained in the Shilov boundary of K. When n = 1, the measure μK is simply harmonic measure for the domain C − K evaluated at infinity. See Section 2. Let F : C → C be a regular polynomial endomorphism; that is, one which extends holomorphically to CP. The filled Jul...

We consider the quadratic family of complex maps given by qc(z) = z 2 + c where c is the center of a hyperbolic component in the Mandelbrot set. Then, we introduce a singular perturbation on the corresponding bounded superattracting cycle by adding one pole to each point in the cycle. When c = −1 the Julia set of q−1 is the well known basilica and the perturbed map is given by fλ(z) = z 2 − 1 +...

In this paper we consider the family of rational maps of the form F λ (z) = z n + λ/z n where n ≥ 2. It is known that there are two cases where the Julia sets of these maps are not connected. If the critical values of F λ lie in the basin of ∞, then the Julia set is a Cantor set. And if the critical values lie in the preimage of the basin surrounding the pole at 0, then the Julia set is a Canto...