Bifurcation measures and quadratic rational maps

We study critical orbits and bifurcations within the moduli space M 2 of quadratic rational maps, f : P 1 → P 1 . We focus on the family of curves, Per 1 ( λ ) ⊂ M 2 for λ ∈ C , defined by the condition that each f ∈ Per 1 ( λ ) has a fixed point of multiplier λ. We prove that the curve Per 1 ( λ ) contains infinitely many postcritically finite maps if and only if λ = 0 , addressing a special case of Baker and DeMarco (‘Special curves and postcritically finite polynomials’, Forum of Math. Pi 1 (2013), doi: 10.1017/fmp.2013.2; Conjecture 1.4). We also show that the two critical points of f define distinct bifurcation measures along Per 1 ( λ ) .