arXiv subjects

Kathryn A. Lindsey

Publications and source records attributed to Kathryn A. Lindsey.

5 recordsLinked to original sources

arXiv2019

Degree-d-invariant laminations

Degree-$d$-invariant laminations of the disk model the dynamical action of a degree-$d$ polynomial; such a lamination defines an equivalence relation on $S^1$ that corresponds to dynamical rays of an associated polynomial landing at the same multi-accessible points in the Julia set. Primitive majors are certain subsets of degree-$d$-invariant laminations consisting of critical leaves and gaps. The space $\textrm{PM}(d)$ of primitive degree-$d$ majors is a spine for the set of monic degree-$d$ polynomials with distinct roots and serves as a parameterization of a subset of the boundary of the connectedness locus for degree-$d$ polynomials. The core entropy of a postcritically finite polynomial is the topological entropy of the action of the polynomial on the associated Hubbard tree. Core entropy may be computed directly, bypassing the Hubbard tree, using a combinatorial analogue of the Hubbard tree within the context of degree-$d$-invariant laminations.

math.DS↗

arXiv2017

Fekete polynomials and shapes of Julia sets

We prove that a nonempty, proper subset $S$ of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if $S$ is bounded and $\hat{\mathbb{C}} \setminus \textrm{int}(S)$ is connected. The proof that such a set is approximable by filled Julia sets is constructive and relies on Fekete polynomials. Illustrative examples are presented. We also prove an estimate for the rate of approximation in terms of geometric and potential theoretic quantities.

math.CV↗

arXiv2017

A Game of Life on Penrose tilings

We define rules for cellular automata played on quasiperiodic tilings of the plane arising from the multigrid method in such a way that these cellular automata are isomorphic to Conway's Game of Life. Although these tilings are nonperiodic, determining the next state of each tile is a local computation, requiring only knowledge of the local structure of the tiling and the states of finitely many nearby tiles. As an example, we show a version of a "glider" moving through a region of a Penrose tiling. This constitutes a potential theoretical framework for a method of executing computations in non-periodically structured substrates such as quasicrystals.

nlin.CG↗

arXiv2016

Shapes of polynomial Julia sets, revisited

Any finite union of disjoint, mutually exterior Jordan curves in the complex plane can be approximated arbitrarily well in the Hausdorff topology by polynomial Julia sets. Furthermore, the proof is constructive.

math.DS↗

arXiv2013

Shapes of Polynomial Julia Sets

Any Jordan curve in the complex plane can be approximated arbitrarily well in the Hausdorff topology by Julia sets of polynomials. Finite collections of disjoint Jordan domains can be approximated by the basins of attraction of rational maps.

math.DS↗