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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
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.
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- Copyright © Cambridge University Press 1994
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