Bifurcation measures and quadratic rational maps

Abstract

We study critical orbits and bifurcations within the moduli space of quadratic rational maps on ${\mathbb{P}}^1$. We focus on the family of curves, $Pe{r}_1(\lambda )$ for $\lambda$ in $\mathbb{C}$, defined by the condition that each $f\in Pe{r}_1(\lambda )$ has a fixed point of multiplier $\lambda$. We prove that the curve $Pe{r}_1(\lambda )$ contains infinitely many postcritically-finite maps if and only if $\lambda =0$; addressing a special case of [BD2, Conjecture 1.4]. We also show that the two critical points of a map $f$ define distinct bifurcation measures along $Pe{r}_1(\lambda )$.

Final version, to appear in Proceedings of the LMS

References in corpus (4)

• Preperiodic points for families of rational map
• On the geometry of bifurcation currents for quadratic rational maps
• Bifurcations, intersections, and heights
• Quasi-adelic measures and equidistribution on ${\mathbb{P}}^1$

Cited by in corpus (5)

• Bifurcations, intersections, and heights
• Quasi-adelic measures and equidistribution on ${\mathbb{P}}^1$
• KAWA 2015: Dynamical moduli spaces and elliptic curves
• Simultaneously preperiodic points for a family of polynomials in positive characteristic
• Bounded geometry for PCF-special subvarieties