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

• Autorzy: Sebastian van Strien
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations
• 37G99: None of the above, but in this section
• 32H50: Iteration problems
• 37E05: Maps of the interval (piecewise continuous, continuous, smooth)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2000
• Tom: 163
• Numer: 1
• Strony: 39-54

Daty

wydano

2000

otrzymano

1999-04-21

poprawiono

1999-09-23

Twórcy

autor

Sebastian van Strien

• Department of Mathematics, University of Warwick, Coventry CV4 7AL, United Kingdom,

Bibliografia

• [AGLV] V. I. Arnol'd, V. V. Goryunov, O. V. Lyashko and V. A. Vasil'ev, Singularity Theory I, Springer, 1998.
• [DH] A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Ecole Norm. Sup. 18 (1985), 287-343.
• [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.
• [TL] L. Tan, Similarity between the Mandelbrot set and Julia sets, Comm. Math. Phys. 134 (1990), 587-617.
• [Tsu] M. Tsujii, A simple proof for monotonicity of entropy in the quadratic family, Ergodic Theory Dynam. Systems, to appear.