Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T16:58:46.408Z Has data issue: false hasContentIssue false
  • English
  • Français

Metric properties of non-renormalizable S-unimodal maps. Part I: Induced expansion and invariant measures

Published online by Cambridge University Press:  19 September 2008

Michael Jakobson  and
Grzegorz Światek
Michael Jakobson
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
Grzegorz Światek
Affiliation:
Mathematics Department, Princeton University, Princeton, NJ 08544, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

For an arbitrary non-renormalizable unimodal map of the interval, f: I → I, with negative Schwarzian derivative, we construct a related map F defined on a countable union of intervals Δ. For each interval Δ, F restricted to Δ is a diffeomorphism which coincides with some iterate of f and whose range is a fixed subinterval of I. If F satisfies conditions of the Folklore Theorem, we call f expansion inducing. Let c be a critical point of f. For f satisfying f″(c) ≠ 0, we give sufficient conditions for expansion inducing. One of the consequences of expansion inducing is that Milnor's conjecture holds for f: the ω-limit set of Lebesgue almost every point is the interval [f2, f(c)]. An important step in the proof is a starting condition in the box case: if for initial boxes the ratio of their sizes is small enough, then subsequent ratios decrease at least exponentially fast and expansion inducing follows.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 4 , December 1994 , pp. 721 - 755
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[BH]Branner, B. and Hubbard, J.. The iteration of cubic polynomials, Part II: patterns and parapatterns. Acta Math. 169 (1992), 229–325.CrossRefGoogle Scholar
[BL]Blokh, A. M. and Lyubich, M. Yu.. Attractors of transformations of an interval. Function. Anal. Appl. 21 (1987), 148–150.Google Scholar
[DH]Douady, A. and Hubbard, J.. On the dynamics of polynomial-like mappings. Ann. Sci. Ec. Sup. 18 (1985), 287–343.Google Scholar
[GJ]Guckenheimer, J. and Johnson, S.. Distortion of S-unimodal maps. Ann. Math. 132 (1990), 71–130.CrossRefGoogle Scholar
[HK]Hofbauer, F. and Keller, G.. Quadratic maps without asymptotic measures. Commun. Math. Phys. 127 (1990), 319–337.CrossRefGoogle Scholar
[J]Jakobson, M.. Absolutely continuous invariant measures for one-parameter families of one dimensional maps. Commun. Math. Phys. 81 (1981), 39–88.CrossRefGoogle Scholar
[J1]Jakobson, M.. Quasisymmetric conjugacy for some one-dimensional maps inducing expansion. Contemp. Math. 135 (1992), 203–211.CrossRefGoogle Scholar
[JS]Jakobson, M. and Światek, G.. Metric properties of non-renormalizable S-unimodal maps. II. Quasisymmetric conjugacy classes. Ergod. Th. & Dynam. Sys. To appear.Google Scholar
[Jo]Johnson, S.. Singular measures without restrictive intervals. Commun. Math. Phys. 110 (1987), 185–190.CrossRefGoogle Scholar
[K]Keller, G.. Exponents, attractors, and Hopf decompositions for interval maps. Ergodic Th. & Dynam. Sys. 10 (1990), 717–744.CrossRefGoogle Scholar
[KN]Keller, G. and Nowicki, T.. Fibonacci maps revisited. Preprint. (1992).Google Scholar
[LM]Lyubich, M and Milnor, J.. The Fibonacci unimodal map. J. Amer. Math. Soc. 6 (1993), 425–457.CrossRefGoogle Scholar
[Ma]Martens, Marco. Distortion results and invariant Cantor sets of unimodal maps. Ergod. Th. & Dynam. Sys. 14 (1994), 331–349.CrossRefGoogle Scholar
[Mi]Milnor, J.. On the concept of an attractor. Commun. Math. Phys. 99 (1985), 177–195.CrossRefGoogle Scholar
[Mis]Misiurewicz, M.. Absolutely continuous invariant measures for certain maps of an interval. Publ. IHES 53 (1981), 17–51.Google Scholar
[MS]Melo, W. de and Strien, S. van. One Dimensional Dynamics. Springer: Berlin, 1993.CrossRefGoogle Scholar
[NS]Nowicki, T. and van Strien, S.. Invariant measures exist under a summability condition for unimodal maps. Inv. Math. 105 (1991), 123–136.CrossRefGoogle Scholar
[Su]Sullivan, D.. Bounds, quadratic differentials and renormalization conjectures, in American Mathematical Society Centennial Publications. Vol. 2, ed Browder, E.. AMS: New York, 1992.Google Scholar
[Sw]Światek, G.. Hyperbolicity is dense in the real quadratic family. Manuscript, version of 18 05, 1993.Google Scholar
[Y]Yoccoz, J. C.. On local connectivity of the Mandelbrot set. To appear.Google Scholar