Let Q be the unit square in the plane and h: Q → h(Q) a quasiconformal map. When h is conformal off a certain self-similar set, the modulus of h(Q) is bounded independent of h. We apply this observation to give explicit estimates for the variation of multipliers of repelling fixed points under a "spinning" quasiconformal deformation of a particular cubic polynomial.
We give an existence theorem for bounded solutions to a system of second order ODEs. Dynamical applications are considered.
We prove real bounds for interval maps with one reflecting critical point.
Let be a degree d polynomial. We say f is post-critically bounded, or PCB, if all of its critical points have bounded orbit under iteration of f. It is known that if p ≥ d and f is PCB, then all critical points of f have p-adic absolute value less than or equal to 1. We give a similar result for 1/2d ≤ p < d. We also explore a one-parameter family of cubic polynomials over ℚ₂ to illustrate that the p-adic Mandelbrot set can be quite complicated when p < d, in contrast with the simple and...
Laminations are classic sets of disjoint and non-self-crossing curves on surfaces. Lamination languages are languages of two-way infinite words which code laminations by using associated labeled embedded graphs, and which are subshifts. Here, we characterize the possible exact affine factor complexities of these languages through bouquets of circles, i.e. graphs made of one vertex, as representative coding graphs. We also show how to build families of laminations together with corresponding lamination...
We construct branched coverings such as matings and captures to describe the dynamics of every critically finite cubic Newton map. This gives a combinatorial model of the set of cubic Newton maps as the gluing of a subset of cubic polynomials with a part of the filled Julia set of a specific polynomial (Figure 1).
Knowledge about the behavior of discontinuous piecewise-linear maps is important for a wide range of applications. An efficient way to investigate the bifurcation structure in 2D parameter spaces of such maps is to detect specific codimension-2 bifurcation points, called organizing centers, and to describe the bifurcation structure in their neighborhood. In this work, we present the organizing centers in the 1D discontinuous piecewise-linear map...
Given a holomorphic mapping of degree we give sufficient conditions on a positive closed (1,1) current of of unit mass under which converges to the Green current as . We also conjecture necessary condition for the same convergence.