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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
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
Grzegorz Światek
Affiliation:
Mathematics Department, Princeton University, Princeton, NJ 08544, USA

Abstract

For an arbitrary non-renormalizable unimodal map of the interval, f: II, 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
Copyright
Copyright © Cambridge University Press 1994

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