Research+Notes

Global Combinatorics of Per_n(0) curves

We introduce a program for showing the Per_n(0) curves (consisting of quadratic rational maps with a marked periodic n-cycle) are irreducible, and prove Per_n(0) is irreducible provided that mating with fixed maps is continuous along veins. Notably, we give an account of periodic polynomial captures (first introduced by Wittner in his thesis, '88) which generalizes polynomial matings, and show several symmetries of the capture operation. The upshot is that the locus of polynomial matings and captures organizes maps nearby a polynomial f, and indicates how to navigate Per_n(0).

Early draft available below. Example of graph G_5 embedded in Per_5 right, suggesting a cell structure for the closure of Per_5(0

combo-connecivity-v4.pdf

Singular Parabolic Implosion (with X. Buff, A. Kapiamba)

We generalize an account of parabolic implosion which describes the local dynamics nearby parabolic parameters to apply nearby the punctures of Per_n(0). Notably, we allow for a pole also to collide with two colliding fixed points, as long as they satisfy certain natural estimates which are satisfy in Per_n(0). One upshot recovers a result of Stimson, Kiwi which states every puncture in Per_n(0) is on the boundary of a polynomial hyperbolic component. Another furnishes an asymptotic local description of the bifurcation locus near punctures, similar to why there are elephants in the Mandelbrot set.

A cell structure for marked cycle curves (with M. Mukundan, D. Stoll, G. Tiozzo)

We introduce a class of marked cycle curves, Cyc_p(F) over a family of rational maps F. For F=Per_1(0) or Per_2(0), we use the intrinsic combinatorics (eg of the Mandelbrot set) to construct algorithmically a cell structure for each Cyc_p(F).

Elastic Graphs for Main Molecule Matings: (D-Powell-Winarski-Yang) http://arxiv.org/abs/2010.11382

Left: Michigan Dynamics Seminar 2/16/21

Right: Annotated Slides

PC-Mating-slides.pdf

PC-Mating-slides(annotated).pdf

An Algorithm to Determine Whether a Mating is a Carpet: (with I. Park, ask for manuscript)

Insung Park and I prove a characterization of when polynomial matings are carpets. Future work would like to use this to quantify the likelihood of matings being carpets.

above: mating loci of f=14/31 and f'=19/31. 14/31 is the first instance of a map satisfying CP(f)=1, meaning "almost every mating with 14/31 is a carpet." No other map polynomial f with fewer than 6 post-critical points has this property. We conjecture that as n gets large, this is the generic case. The high carpet probability is reflected in parameter space by being surrounded by a"fuzzy" region. Contrast this with the mating locus of 19/31 which has an entire limb's worth of parameters that correspond to non-carpet matings. Shown between pink and green.

carpet-video.mp4

above: video courtesty of Cheritat of a minimal (in terms of post-critical set size) possible Sierpinski Carpet mating, with 8=3+5 post-critical points; the mating of the airplane and the kokopelli in the 1/4-limb with crtiical value 7/31.