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

top
We show that in normalized families of polynomial or rational maps, Misiurewicz maps (critically finite or infinite) unfold generically. For example, if is critically finite with non-degenerate critical point such that are hyperbolic periodic points for i = 1,...,n, then  IV-1. Age impartible......................................................................................................................................................................... 31   is a local diffeomorphism for λ near . For quadratic families this result was proved previously in DH using entirely different methods.

How to cite

top

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

top
  1. [AGLV] V. I. Arnol'd, V. V. Goryunov, O. V. Lyashko and V. A. Vasil'ev, Singularity Theory I, Springer, 1998. 
  2. [DH] A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Ecole Norm. Sup. 18 (1985), 287-343. Zbl0587.30028
  3. [Le] O. Lehto, Univalent Functions and Teichmüller Spaces, Grad. Texts in Math. 109, Springer, 1987. 
  4. [LV] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer, 1973. Zbl0267.30016
  5. [LS1] G. Levin and S. van Strien, Local connectivity of the Julia set of real polynomials, Ann. of Math. 147 (1998), 471-541. Zbl0941.37031
  6. [LS2] G. Levin and S. van Strien, Total disconnectedness of the Julia set of real polynomials, Astérisque, to appear. 
  7. [Ma1] R. Mañé, Hyperbolicity, sinks and measure in one dimensional dynamics, Comm. Math. Phys. 100 (1985), 495-524. Zbl0583.58016
  8. [Ma2] R. Mañé, On a theorem of Fatou, Bol. Soc. Brasil. Mat. 24 (1993), 1-11. Zbl0781.30023
  9. [MSS] R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. Ecole Norm. Sup. 16 (1983), 193-217. Zbl0524.58025
  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
  13. [TL] L. Tan, Similarity between the Mandelbrot set and Julia sets, Comm. Math. Phys. 134 (1990), 587-617. Zbl0726.58026
  14. [Tsu] M. Tsujii, A simple proof for monotonicity of entropy in the quadratic family, Ergodic Theory Dynam. Systems, to appear. 

NotesEmbed ?

top

You must be logged in to post comments.