Distribution of polynomials with cycles of given multiplier

TL;DR

This paper investigates how polynomials of degree d with cycles of a specific period and multiplier distribute within the parameter space, revealing a balanced bifurcation current across these conditions.

Contribution

It demonstrates that hypersurfaces defined by cycles with given period and multiplier equidistribute the bifurcation current in polynomial families.

Findings

Hypersurfaces are equidistributed with respect to the bifurcation current.

Distribution is uniform across cycles of fixed period and multiplier.

Provides a new understanding of the parameter space structure for polynomial dynamics.

Abstract

In the family of degree $d$ polynomials the hypersurfaces defined by the existence of a cycle of period $n$ and multiplier $e^{i\theta}$ are shown to equiditribute the bifurcation current.