Geodesic
Misiurewicz maps unfold generically (even if they are critically non-finite)
Fundamenta Mathematicae, Tome 163 (2000) no. 1, pp. 39-54.

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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 $f_{λ_0}$ is critically finite with non-degenerate critical point $c_1(λ_0),...,c_n(λ_0)$ such that $f_{λ_0}^{k_i}(c_i(λ_0)) = p_i(λ_0)$ are hyperbolic periodic points for i = 1,...,n, then  $$λ ↦ (f_λ^{k_1}(c_1(λ))-p_1(λ),..., f_λ^{k_{d-2}}(c_{d-2}(λ))-p_{d-2}(λ))$$ is a local diffeomorphism for λ near $λ_0$. For quadratic families this result was proved previously in {DH} using entirely different methods.
DOI : 10.4064/fm-163-1-39-54

Sebastian van Strien 1

1 
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Sebastian van Strien. Misiurewicz maps unfold generically (even if they are critically non-finite). Fundamenta Mathematicae, Tome 163 (2000) no. 1, pp. 39-54. doi : 10.4064/fm-163-1-39-54. http://geodesic.mathdoc.fr/articles/10.4064/fm-163-1-39-54/

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