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We show that in normalized families of polynomial or rational maps, Misiurewicz maps (critically finite or infinite) unfold generically. For example, if fλ0 is critically finite with non-degenerate critical point c1(λ0), . . . , cn(λ0) such that f ki λ0 (ci(λ0)) = pi(λ0) are hyperbolic periodic points for i = 1, . . . , n, then λ 7→ (f1 λ (c1(λ))− p1(λ), . . . , f kd−2 λ (cd−2(λ))− pd−2(λ)) is a local diffeomorphism for λ near λ0. For quadratic families this result was proved previously in [DH] using entirely different methods.

Misiurewicz maps unfold generically (even if they are critically non-finite)