Topological invariance of the Collet–Eckmann property for S-unimodal maps

Tomasz Nowicki; Feliks Przytycki

Fundamenta Mathematicae (1998)

  • Volume: 155, Issue: 1, page 33-43
  • ISSN: 0016-2736

Abstract

top
We prove that if f, g are smooth unimodal maps of the interval with negative Schwarzian derivative, conjugated by a homeomorphism of the interval, and f is Collet-Eckmann, then so is g.

How to cite

top

Nowicki, Tomasz, and Przytycki, Feliks. "Topological invariance of the Collet–Eckmann property for S-unimodal maps." Fundamenta Mathematicae 155.1 (1998): 33-43. <http://eudml.org/doc/212241>.

@article{Nowicki1998,
abstract = {We prove that if f, g are smooth unimodal maps of the interval with negative Schwarzian derivative, conjugated by a homeomorphism of the interval, and f is Collet-Eckmann, then so is g.},
author = {Nowicki, Tomasz, Przytycki, Feliks},
journal = {Fundamenta Mathematicae},
keywords = {-unimodal maps; absolutely continuous invariant measure; Collet-Eckmann condition; topological invariants; topological conjugacy; holomorphic dynamics},
language = {eng},
number = {1},
pages = {33-43},
title = {Topological invariance of the Collet–Eckmann property for S-unimodal maps},
url = {http://eudml.org/doc/212241},
volume = {155},
year = {1998},
}

TY - JOUR
AU - Nowicki, Tomasz
AU - Przytycki, Feliks
TI - Topological invariance of the Collet–Eckmann property for S-unimodal maps
JO - Fundamenta Mathematicae
PY - 1998
VL - 155
IS - 1
SP - 33
EP - 43
AB - We prove that if f, g are smooth unimodal maps of the interval with negative Schwarzian derivative, conjugated by a homeomorphism of the interval, and f is Collet-Eckmann, then so is g.
LA - eng
KW - -unimodal maps; absolutely continuous invariant measure; Collet-Eckmann condition; topological invariants; topological conjugacy; holomorphic dynamics
UR - http://eudml.org/doc/212241
ER -

References

top
  1. [B1] H. Bruin, Invariant measures of interval maps, PhD thesis, Tech. Univ. Delft, 1994. 
  2. [B2] H. Bruin, Topological conditions for the existence of invariant measures for unimodal maps, Ergodic Theory Dynam. Systems 14 (1994), 433-452. 
  3. [CE] P. Collet and J.-P. Eckmann, Positive Lyapunov exponents and absolute continuity for maps of the interval, ibid. 3 (1983), 13-46. Zbl0532.28014
  4. [CJY] L. Carleson, P. Jones and J.-C. Yoccoz, Julia and John, Bol. Soc. Brasil. Mat. 25 (1994), 1-30. 
  5. [DPU] M. Denker, F. Przytycki and M. Urbański, On the transfer operator for rational functions on the Riemann sphere, Ergodic Theory Dynam. Systems 16 (1996), 255-266. Zbl0852.46024
  6. [GS] J. Graczyk and S. Smirnov, Collet, Eckmann, & Hölder, Invent. Math., to appear. 
  7. [JS] M. Jakobson and G. Świątek, Metric properties of non-renormalizable S-unimodal maps, II. Quasisymmetric conjugacy classes, Ergodic Theory Dynam. Systems 15 (1995), 871-938. Zbl0920.54037
  8. [M] R. Ma né, On a theorem of Fatou, Bol. Soc. Brasil. Mat. 24 (1993), 1-12. 
  9. [MS] W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer, 1993. Zbl0791.58003
  10. [NP] T. Nowicki and F. Przytycki, The conjugacy of Collet-Eckmann's map of the interval with the tent map is Hölder continuous, Ergodic Theory Dynam. Systems 9 (1989), 379-388. Zbl0664.58014
  11. [NS] T. Nowicki and D. Sands, Nonuniform hyperbolicity and universal bounds for S-unimodal maps, Invent. Math., to appear. Zbl0908.58016
  12. [P1] F. Przytycki, Iterations of holomorphic Collet-Eckmann maps, conformal and invariant measures, Trans. Amer. Math. Soc., to appear. Zbl0892.58063
  13. [P2] F. Przytycki, On measure and Hausdorff dimension of Julia sets for holomorphic Collet-Eckmann maps, in: International Conference on Dynamical Systems, Montevideo 1995 - a Tribute to Ricardo Ma né (F. Ledrappier, J. Lewowicz and S. Newhouse, eds.), Pitman Res. Notes Math. Ser. 362, Longman, 1996, 167-181. 
  14. [P3] F. Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc. 119 (1993), 309-317. Zbl0787.58037
  15. [P4] F. Przytycki, Hölder implies CE, Astérisque, volume dedicated to A. Douady on his 60th birthday, to appear. 
  16. [PR1] F. Przytycki and S. Rohde, Porosity of Collet-Eckmann Julia sets, Fund. Math., to appear. Zbl0908.58054
  17. [PR2] F. Przytycki and S. Rohde, Rigidity of holomorphic Collet-Eckmann repellers, preprint, May 1997. Zbl1034.37026
  18. [S] D. Sands, Topological conditions for positive Lyapunov exponent in unimodal case, Ph.D. thesis, St. John's College, Cambridge, 1995. 
  19. [SN] D. Sands and T. Nowicki, Quasisymmetric conjugacies of Collet-Eckmann maps, Ergodic Theory Dynam. Systems, to appear. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.