English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

By studying laminations of the unit disk, we can gain insight into thestructure of Julia sets of polynomials and their dynamics in the complex plane.The polynomials of a given degree, $d$, have a parameter space. The hyperboliccomponents of such parameter spaces are in correspondence to rotationalpolygons, or classes of "rotational sets", which we study in this paper. Bystudying the count of such rotational sets, and therefore the underlyingstructure behind these rotational sets and polygons, we can gain insight intothe interrelationship among hyperbolic components of the parameter space ofthese polynomials.  These rotational sets are created by uniting rotational orbits, as we definein this paper. The number of such sets for a given degree $d$, rotation number$\frac pq$, and cardinality $k$ can be determined by analyzing the potentialplacements of pre-images of zero on the unit circle with respect to therotational set under the $d$-tupling map. We obtain a closed-form formula forthe count. Though this count is already known based upon some sophisticatedresults, our count is based upon elementary geometric and combinatorialprinciples, and provides an intuitive explanation.

Counting Rotational Sets for Laminations of the Unit Disk from First Principles