We study critical orbits and bifurcations within the moduli space M2 of quadratic rational maps, f : P1 → P1. We focus on the family of curves, Per1(λ) ⊂ M2 for λ ε C, defined by the condition that each f ε Per1(λ) has a fixed point of multiplier λ. We prove that the curve Per1(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 Per1(λ).
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