On backward stability of holomorphic dynamical systems

Genadi. Levin

Fundamenta Mathematicae (1998)

  • Volume: 158, Issue: 2, page 97-107
  • ISSN: 0016-2736

Abstract

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For a polynomial with one critical point (maybe multiple), which does not have attracting or neutral periodic orbits, we prove that the backward dynamics is stable provided the Julia set is locally connected. The latter is proved to be equivalent to the non-existence of a wandering continuum in the Julia set or to the shrinking of Yoccoz puzzle-pieces to points.

How to cite

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Levin, Genadi.. "On backward stability of holomorphic dynamical systems." Fundamenta Mathematicae 158.2 (1998): 97-107. <http://eudml.org/doc/212311>.

@article{Levin1998,
abstract = {For a polynomial with one critical point (maybe multiple), which does not have attracting or neutral periodic orbits, we prove that the backward dynamics is stable provided the Julia set is locally connected. The latter is proved to be equivalent to the non-existence of a wandering continuum in the Julia set or to the shrinking of Yoccoz puzzle-pieces to points.},
author = {Levin, Genadi.},
journal = {Fundamenta Mathematicae},
keywords = {critical point; attracting periodic orbits; neutral periodic orbits; Julia set; locally connected; wandering continuum; Yoccoz puzzle-pieces},
language = {eng},
number = {2},
pages = {97-107},
title = {On backward stability of holomorphic dynamical systems},
url = {http://eudml.org/doc/212311},
volume = {158},
year = {1998},
}

TY - JOUR
AU - Levin, Genadi.
TI - On backward stability of holomorphic dynamical systems
JO - Fundamenta Mathematicae
PY - 1998
VL - 158
IS - 2
SP - 97
EP - 107
AB - For a polynomial with one critical point (maybe multiple), which does not have attracting or neutral periodic orbits, we prove that the backward dynamics is stable provided the Julia set is locally connected. The latter is proved to be equivalent to the non-existence of a wandering continuum in the Julia set or to the shrinking of Yoccoz puzzle-pieces to points.
LA - eng
KW - critical point; attracting periodic orbits; neutral periodic orbits; Julia set; locally connected; wandering continuum; Yoccoz puzzle-pieces
UR - http://eudml.org/doc/212311
ER -

References

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  5. [H] J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, preprint, IHES/M/92/79 (1992). Zbl0797.58049
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  7. [LvS] G. Levin and S. van Strien, Local connectivity of the Julia set of real polynomials, Ann. of Math. 147 (1998), 471-541. Zbl0941.37031
  8. [Ly] M. Lyubich, Geometry of quadratic polynomials: moduli, rigidity and local connectivity, Acta Math. 178 (1997), 185-297. 
  9. [Ma] R. Mañé, On a theorem of Fatou, Bol. Soc. Brasil. Math. 24 (1993), 1-11. Zbl0781.30023
  10. [McM] C. McMullen, Complex Dynamics and Renormalization, Ann. of Math. Stud. 135, Princeton Univ. Press, 1994. 
  11. [Mi1] J. Milnor, Dynamics in one complex variable: introductory lectures, Stony Brook Preprint 1990/5. 
  12. [Mi2] J. Milnor, Local connectivity of Julia sets: expository lectures, Stony Brook Preprint 1992/11. 
  13. [Pr] F. Przytycki, Iteration of holomorphic Collet-Eckmann maps: conformal and invariant measures, Trans. Amer. Math. Soc. 352 (1998), 717-742. Zbl0892.58063
  14. [P-M] R. Pérez-Marco, Topology of Julia sets and hedgehogs, Orsay, preprint 94-48 (1994). 
  15. [R] P. Roesch, thesis, Lyon, 1997. 
  16. [Th] W. Thurston, The combinatorics of iterated rational maps, preprint, 1984. 
  17. [Y] J.-C. Yoccoz, Sur la connectivité locale de M, 1989. 

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