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Metric properties of non-renormalizable S-unimodal maps: II. Quasisymmetric conjugacy classes

Published online by Cambridge University Press:  14 October 2010

M. Jakobson  and
G. Świạtek
M. Jakobson
Affiliation:
Mathematics Department, University of Maryland, College Park, MD 20742, USA†
G. Świạtek
Affiliation:
Mathematics Department, Penn State University, Mc Allister 209, University Park, PA 16802, USA‡
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Abstract

It is shown that for certain classes of S-unimodal maps with aperiodic kneading sequences, the topological conjugacies are also quasisymmetric. This includes some infinitely renormalizable polynomials of unbounded type.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 5 , October 1995 , pp. 871 - 938
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

[1] Ahlfors, L. V.. Quasiconformal reflections. Ada Math. 109 (1963), 291–301.Google Scholar
[2] Benedicks, M. and Carleson, L.. The dynamics of the Hgnon map. Ann. Math. 133 (1991), 73–169.CrossRefGoogle Scholar
[3] Beurling, A. and Ahlfors, L.. The boundary correspondence under quasiconformal mappings. Ada Math. 96 (1956), 125–142.Google Scholar
[4] Blokh, A. M. and Lyubich, M. Yu. Measure and dimension of solenoidal attractors of one-dimensional dynamical systems. Commun. Math. Phys. 127 (1990), 573–583.CrossRefGoogle Scholar
[5] Collet, P. and Eckmann, J. P.. Positive Liapunov exponents and absolute continuity for maps of the interval. Ergod. Th. & Dynam. Sys. 3 (1983), 13–46.CrossRefGoogle Scholar
[6] Douady, A. and Hubbard, J. H.. Etude dynamique des polynfimes complexes. Publ. Math. d'Orsay 84–82 (1984).Google Scholar
[7] Melo, W. De and Strien, S. Van. One-dimensional Dynamics. Springer, New York-Berlin-Heidelberg, 1993.CrossRefGoogle Scholar
[8] Douady, A. and Hubbard, J. H.. On the dynamics of polynomial-like mappings. Ann. Sci. Ec. Norm. Sup. (Paris) 18 (1985), 287–343.Google Scholar
[9] Gehring, F.. Charaderistic properties of quasidiscs. Presses de l'Université de Montréal, Montreal, 1982.Google Scholar
[10] Graczyk, J. and Swiatek, G.. Critical circle maps near bifurcation, Stony Brook IMS. Preprint. 1991.Google Scholar
[11] Graczyk, J. and Swiatek, G.. Induced expansion for quadratic polynomials. Preprint. Stony Brook 8 (1993).Google Scholar
[12] Guckenheimer, J.. Sensitive dependence on initial conditions for one dimensional maps. Commun. Math. Phys. 70 (1979), 133–160.CrossRefGoogle Scholar
[13] Guckenheimer, J.. Limit sets of S-unimodal maps with zero entropy. Commun. Math. Phys. 110 (1987), 655–659.CrossRefGoogle Scholar
[14] Guckenheimer, J. and Johnson, S.. Distortion of Sunimodal maps. Ann. Math. 132 (1990), 71–130.CrossRefGoogle Scholar
[15] Hubbard, J. H.. According to Yoccoz, J. C.: Puzzles and quadratic tableaux, manuscript.Google Scholar
[16] Jakobson, M.. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81 (1981), 39–88.CrossRefGoogle Scholar
[17] Jakobson, M.. Quasisymmetric conjugacies for some one-dimensional maps inducing expansion. Contemp. Math. 135 (1992), 203–211.CrossRefGoogle Scholar
[18] Jakobson, M. and Swiatek, G.. Metric properties of non-renormalizable S-unimodal maps. Ergod. Tli. & Dynam. Sys. 14 (1994), 721–756.CrossRefGoogle Scholar
[19] Lento, O. and Virtanen, K.. Quasikonfonne Abbildungen. Springer. Berlin-Heidelberg-New York, 1965.Google Scholar
[20] Mane, R., Sad, P. and Sullivan, D.. On the dynamics of rational maps. Ann. Ec. Norm. Sup. 16 (1983), 193–217.CrossRefGoogle Scholar
[21] Metropolis, N., Stein, M. L.Stein, P. R.. On finite limit sets for transformations on the unit interval. J. Combin. Theory (A)15 (1973), 25–44.CrossRefGoogle Scholar
[22] Milnor, J.Thurston, W.. On iterated maps of the interval I, II. Springer Lecture Notes in Mathematics 1342. Springer, Berlin, 1988.Google Scholar
[23] Nowicki, T.. Symmetric S-unimodal mappings and positive Lyapunov exponent. Ergod. Th. & Dynam. Sys. 5 (1985), 611–616.CrossRefGoogle Scholar
[24] Preston, C.. Iterates of Maps on an Interval. Springer Lecture Notes in Mathematics, 999. Springer, Berlin-Heidelberg-New York, 1983.CrossRefGoogle Scholar
[25] Sullivan, D.. Quasiconformal homeomorphisms and dynamics. I. Fatou-Julia problem on wandering domains. Ann. Math. 122 (1985), 401–418.CrossRefGoogle Scholar
[26] Sullivan, D.. Bounds, quadratic differentials and renormalization conjectures. American Mathematical Society Centennial Publications. Vol. 2. American Mathematical Society, Providence, RI, 1991.Google Scholar
[27] Thieullen, P., Tresser, C. and Young, L. S.. Exposant de Lyapunov positif dans des families a un paramgtre d'applications unimodales. CR Acad. Sci. Paris Serie 1 315 (1992), 69–72.Google Scholar