Dynamics of quadratic polynomials : complex bounds for real maps

Mikhail Lyubich; Michael Yampolsky

Annales de l'institut Fourier (1997)

  • Volume: 47, Issue: 4, page 1219-1255
  • ISSN: 0373-0956

Abstract

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We prove complex bounds for infinitely renormalizable real quadratic maps with essentially bounded combinatorics. This is the last missing ingredient in the problem of complex bounds for all infinitely renormalizable real quadratics. One of the corollaries is that the Julia set of any real quadratic map z ↦ z 2 + c , c ∈ [ - 2 , 1 / 4 ] , is locally connected.

How to cite

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Lyubich, Mikhail, and Yampolsky, Michael. "Dynamics of quadratic polynomials : complex bounds for real maps." Annales de l'institut Fourier 47.4 (1997): 1219-1255. <http://eudml.org/doc/75261>.

@article{Lyubich1997,
abstract = {We prove complex bounds for infinitely renormalizable real quadratic maps with essentially bounded combinatorics. This is the last missing ingredient in the problem of complex bounds for all infinitely renormalizable real quadratics. One of the corollaries is that the Julia set of any real quadratic map $z\mapsto z^2+c$, $c\in [-2,1/4]$, is locally connected.},
author = {Lyubich, Mikhail, Yampolsky, Michael},
journal = {Annales de l'institut Fourier},
keywords = {one-dimensional dynamics; renormalization; quadratic polynomials; complex bounds; local connectivity},
language = {eng},
number = {4},
pages = {1219-1255},
publisher = {Association des Annales de l'Institut Fourier},
title = {Dynamics of quadratic polynomials : complex bounds for real maps},
url = {http://eudml.org/doc/75261},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Lyubich, Mikhail
AU - Yampolsky, Michael
TI - Dynamics of quadratic polynomials : complex bounds for real maps
JO - Annales de l'institut Fourier
PY - 1997
PB - Association des Annales de l'Institut Fourier
VL - 47
IS - 4
SP - 1219
EP - 1255
AB - We prove complex bounds for infinitely renormalizable real quadratic maps with essentially bounded combinatorics. This is the last missing ingredient in the problem of complex bounds for all infinitely renormalizable real quadratics. One of the corollaries is that the Julia set of any real quadratic map $z\mapsto z^2+c$, $c\in [-2,1/4]$, is locally connected.
LA - eng
KW - one-dimensional dynamics; renormalization; quadratic polynomials; complex bounds; local connectivity
UR - http://eudml.org/doc/75261
ER -

References

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  1. [BL1] A. BLOKH and M. LYUBICH, Measure of solenoidal attractors of unimodal maps of the segment., Math. Notes, 48 (1990), 1085-1990. Zbl0774.58026MR92d:58117
  2. [BL2] A. BLOKH and M. LYUBICH, Measure and dimension of solenoidal attractors of one dimensional dynamical systems, Commun. of Math. Phys., 1 (1990), 573-583. Zbl0721.58033MR91g:58164
  3. [D] A. DOUADY, Chirurgie sur les applications holomorphes, in Proc. ICM, Berkeley, 1986, 724-738. Zbl0698.58048MR89k:58139
  4. [DH] A. DOUADY, J.H. HUBBARD, On the dynamics of polynomial-like maps, Ann. Sci. Éc. Norm. Sup., 18 (1985), 287-343. Zbl0587.30028MR87f:58083
  5. [E] H. EPSTEIN, Fixed points of composition operators, Nonlinearity, 2 (1989), 305-310. Zbl0731.39007MR90j:58086
  6. [F] E. DE FARIA, Proof of universality for critical circle mappings, Thesis, CUNY (1992). 
  7. [G] J. GUCKENHEIMER, Limit sets of S-unimodal maps with zero entropy, Comm. Math. Phys., 110 (1987), 655-659. Zbl0625.58027MR88i:58111
  8. [GJ] J. GUCKENHEIMER, S. JOHNSON, Distortion of S-unimodal maps, Ann. Math., 132 (1990), 71-130. Zbl0708.58007MR91g:58157
  9. [GS1] J. GRACZYK, G. SWIATEK, Induced expansion for quadratic polynomials, Ann. Sci. Éc. Norm. Sup., 29 (1996), 399-482. Zbl0867.58048MR98d:58152
  10. [GS2] J. GRACZYK, G. SWIATEK, Polynomial-like property for real quadratic polynomials, Preprint, 1995. Zbl0843.58075
  11. [H] J.H. HUBBARD, Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, in Topological Methods in Modern Mathematics, A Symposium in Honor of John Milnor's 60th Birthday, Publish or Perish, 1993. Zbl0797.58049
  12. [HJ] J. HU, Y. JIANG, The Julia set of the Feigenbaum quadratic polynomial is locally connected, Preprint, 1993. 
  13. [He] M. HERMAN, Conjugaison quasi-symétrique des homomorphismes analytiques du cercle à des rotations, Preprint. 
  14. [J] Y. JIANG, Infinitely renormalizable quadratic Julia sets, Preprint 1993. Zbl0947.37029
  15. [LS] G. LEVIN, S. VAN STRIEN, Local connectivity of Julia sets of real polynomials, Preprint IMS at Stony Brook, # 1995/5. Zbl0941.37031
  16. [L1] M. LYUBICH, Combinatorics, geometry and attractors of quasi-quadratic maps, Ann. Math, 140 (1994), 347-404. Zbl0821.58014
  17. [L2] M. LYUBICH, Dynamics of quadratic polynomials, I-II, Acta Math, 178 (1997), 185-297. Zbl0908.58053MR98e:58145
  18. [M1] J. MILNOR, Local connectivity of Julia sets: expository lectures, Preprint IMS at Stony Brook #1992/11. Zbl1107.37305
  19. [M2] J. MILNOR, Periodic orbits, external rays and the Mandelbrot set: An expository account, Preprint, 1995. 
  20. [MT] J. MILNOR, W. THURSTON, On iterated maps of the interval, Lecture Notes in Mathematics 1342, Springer Verlag (1988), 465-563. Zbl0664.58015MR90a:58083
  21. [McM1] C. MCMULLEN, Complex dynamics and renormalization, Annals of Math. Studies, 135, Princeton University Press (1994). Zbl0822.30002
  22. [McM2] C. MCMULLEN, Renormalization and 3-manifolds which fiber over the circle, Annals of Math. Studies, Princeton University Press (1996). Zbl0860.58002MR97f:57022
  23. [Ma] M. MARTENS, Distortion results and invariant Cantor sets for unimodal maps, Erg. Th. and Dyn. Syst., 14 (1994), 331-349. Zbl0809.58026MR96c:58108
  24. [MS] W. DE MELO, S. VAN STRIEN, One dimensional dynamics, Springer-Verlag (1993). Zbl0791.58003MR95a:58035
  25. [R] M. REES, A possible approach to a complex renormalization problem, in Linear and Complex Analysis Problem Book 3, part II, pp. 437-440, Lecture Notes in Math., 1574. 
  26. [S] D. SULLIVAN, Bounds, quadratic differentials, and renormalization conjectures, AMS Centennial Publications, 2: Mathematics into Twenty-first Century, 1992. Zbl0936.37016MR93k:58194
  27. [Y] M. YAMPOLSKY, Complex bounds for critical circle maps, Preprint IMS at Stony Brook, # 1995/12. 

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