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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
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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

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