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

Sebastian van Strien

Fundamenta Mathematicae (2000)

  • Volume: 163, Issue: 1, page 39-54
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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van Strien, Sebastian. "Misiurewicz maps unfold generically (even if they are critically non-finite)." Fundamenta Mathematicae 163.1 (2000): 39-54. <http://eudml.org/doc/212428>.

@article{vanStrien2000,
abstract = {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.},
author = {van Strien, Sebastian},
journal = {Fundamenta Mathematicae},
keywords = {conjugacy; rational maps; Misiurewicz map},
language = {eng},
number = {1},
pages = {39-54},
title = {Misiurewicz maps unfold generically (even if they are critically non-finite)},
url = {http://eudml.org/doc/212428},
volume = {163},
year = {2000},
}

TY - JOUR
AU - van Strien, Sebastian
TI - Misiurewicz maps unfold generically (even if they are critically non-finite)
JO - Fundamenta Mathematicae
PY - 2000
VL - 163
IS - 1
SP - 39
EP - 54
AB - 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.
LA - eng
KW - conjugacy; rational maps; Misiurewicz map
UR - http://eudml.org/doc/212428
ER -

References

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  10. [McM] C. McMullen, Complex Dynamics and Renormalization, Ann. of Math. Stud. 135, Princeton Univ. Press, 1994. 
  11. [MS] W. de Melo and S. van Strien, One-Dimensional Dynamics, Ergeb. Math. Grenzgeb. 25, Springer, 1993. Zbl0791.58003
  12. [ST] M. Shishikura and L. Tan, Mañé's theorem, to appear. Zbl1062.37046
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  14. [Tsu] M. Tsujii, A simple proof for monotonicity of entropy in the quadratic family, Ergodic Theory Dynam. Systems, to appear. 

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