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

Sebastian van Strien — 2000

Fundamenta Mathematicae

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 IV-1. Age impartible......................................................................................................................................................................... 31 $λ \mapsto (f_{λ}^{k_{1}} (c_{1} (λ)) - p_{1} (λ), . . ., f_{λ}^{k_{d - 2}} (c_{d - 2} (λ)) - p_{d - 2} (λ))$ is a local diffeomorphism...

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

Invariant measures exist under a summability condition for unimodal maps.

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