Equations at Infinity for Critical-Orbit-Relation Families of Rational Maps

Abstract

We develop techniques for using compactifications of Hurwitz spaces to study families of rational maps ${\mathbb{P}}^1\to {\mathbb{P}}^1$ defined by critical orbit relations. We apply these techniques in two settings: We show that the parameter space ${\text{Per}}_{d,4}$ of degree-d bicritical maps with a marked 4-periodic critical point is a d²-punctured Riemann surface of genus $\frac{(d-1)(d-2)}{2}$. We also recover a result of Canci and Vishkautsan, showing that the parameter space ${\text{Per}}_{2,5}$ of degree-2 rational maps with a marked 5-periodic critical point is a 10-punctured elliptic curve, and identifying its isomorphism class over $\mathbb{Q}$. We carry out an experimental study of the interaction between dynamically defined points of ${\text{Per}}_{2,5}$ (such as PCF points or punctures) and the group structure of the underlying elliptic curve.

Keywords:

• Complex dynamics
• rational maps
• Hurwitz spaces
• moduli spaces

Acknowledgments

The authors thank Sarah Koch and Joseph H. Silverman for useful conversations; the first author also thanks Xavier Buff, Laura DeMarco, and Rob Benedetto. The authors also thank Max Weinreich for bringing [9] to their attention, Caroline Davis for pointing out an error in the position of one puncture in Figure 1, and the anonymous referees for very helpful feedback.

Notes

1 Recall that the gonality of a Riemann surface X is the minimum degree of a nonconstant holomorphic map $X\to {\mathbb{P}}^1$.

2 The fact that a smooth plane curve of degree d has gonality exactly d – 1 is classical, see [30].

3 Note that non-simple critical points are marked by sets of indices; see e.g., [5, p.1].

4 For precise statements on the structure of these constraints, called “postcritical portraits,” see [19, section. 9]; we do not need these to study ${\text{Per}}_{d,n}$.

5 Indeed, we will need to use formal power series in our analysis; see e.g., Section 4.2.6, which involves computing Taylor series expansions of the square root function.

6 It happens that there do exist Zariski coordinates around R₆, making the analysis similar to that of R₅. We chose these coordinates instead to demonstrate the most general form of the process, and why it is necessary to work with power series.

Additional information

Funding

Rohini Ramadas was supported by an NSF postdoctoral fellowship (DMS-1703308), and a Tamarkin Assistant Professorship at Brown University. Rob Silversmith was supported by an RTG-Zelevinsky Research Instructorship at Northeastern University, part of NSF grant DMS-1645877.