The space Per_n is the algebraic subspace of rat_2, defined as the space of quadratic rational maps modulo Mobius transformations with a marked critical cycle of period n. An open problem, due to J. Milnor (in the 1990's), is whether Per_n is irreducible for all integers n. In this talk I will discuss two approaches to Milnor's question. The first (joint with S. Koch) follows work of A, Epstein, where one studies at an analog of Per_n in the Teichmueller space of n+1 marked points on a sphere. The second is work in progress with C. Davis and A. Kapiamba, where Per_n is explored simultaneously as a parameter space with punctures and as an infinite collection of punctured algebraic curves in a complex affine plane.