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

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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

 IV-1. Age impartible......................................................................................................................................................................... 31
 $λ ↦ (f_λ^{k_1}(c_1(λ))-p_1(λ),..., f_λ^{k_{d-2}}(c_{d-2}(λ))-p_{d-2}(λ))$ is a local diffeomorphism for λ near $λ_0$. For quadratic families this result was proved previously in {DH} using entirely different methods.
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  • [AGLV] V. I. Arnol'd, V. V. Goryunov, O. V. Lyashko and V. A. Vasil'ev, Singularity Theory I, Springer, 1998.
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  • [Le] O. Lehto, Univalent Functions and Teichmüller Spaces, Grad. Texts in Math. 109, Springer, 1987.
  • [LV] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer, 1973.
  • [LS1] G. Levin and S. van Strien, Local connectivity of the Julia set of real polynomials, Ann. of Math. 147 (1998), 471-541.
  • [LS2] G. Levin and S. van Strien, Total disconnectedness of the Julia set of real polynomials, Astérisque, to appear.
  • [Ma1] R. Mañé, Hyperbolicity, sinks and measure in one dimensional dynamics, Comm. Math. Phys. 100 (1985), 495-524.
  • [Ma2] R. Mañé, On a theorem of Fatou, Bol. Soc. Brasil. Mat. 24 (1993), 1-11.
  • [MSS] R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. Ecole Norm. Sup. 16 (1983), 193-217.
  • [McM] C. McMullen, Complex Dynamics and Renormalization, Ann. of Math. Stud. 135, Princeton Univ. Press, 1994.
  • [MS] W. de Melo and S. van Strien, One-Dimensional Dynamics, Ergeb. Math. Grenzgeb. 25, Springer, 1993.
  • [ST] M. Shishikura and L. Tan, Mañé's theorem, to appear.
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bwmeta1.element.bwnjournal-article-fmv163i1p39bwm
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