Measure of minimal sets of polymodal maps

Vargas, Edson

doi:10.1017/s0143385700008750

“…Real bounds as in Theorem A, but around recurrent turning points, were proved previously by Martens in the negative Schwarzian unimodal case and by Vargas in the case of C 2 multimodal maps without inflection points; see [14], [23] and also Shen's paper [18]. If all branches of f are monotone and there is at most one critical point (of inflection type), then such bounds were proved by Levin; see [8] and also [10].…”

Section: Corollary Of the Proof Of Theorem A (Non-existence Of Wandersupporting

confidence: 57%

Untitled

2004

J. Amer. Math. Soc.

0

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SEBASTIAN VAN STRIEN AND EDSON VARGAS i.e., the points where f is zero. Throughout this paper we will assume that f is C k outside these critical points, and that f is non-flat at these critical point, i.e., that for i = 1,. .. , d and x near c i we can write f (x) = ±|φ i (x)| βi + f (c i), where φ i is C k , φ i (c i) = 0 and β i > 1. Here k = 2 is enough for Theorems A, B(1) and C(1), and k = 3 is enough for the remaining theorems. In fact, for Theorems A, B(1) and C(1) it suffices to take k = 1 + Zygmund; see [17]. We denote the class of such maps by A k and note that A 3 ⊂ A 2 ⊂ A 1+Zygmund. If B is a Borelean set, we will denote its Lebesgue measure by |B|. We will also use the following two definitions: Definition 1. Let U, V be bounded intervals such that the closure of U is contained in the interior of V. We say that V is an α-scaled neighbourhood of U if |U + | ≥ α|U | and |U − | ≥ α|U | , where U + and U − are the connected components of V \ U. We also sometimes say that U is α-well-inside V. Definition 2. An open interval I ⊂ [−1, 1] is called a nice interval if the forward orbit of its boundary does not intersect I; that is, I ∩ f i (∂I) = ∅ for each i ≥ 0.

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“…Real bounds as in Theorem A, but around recurrent turning points, were proved previously by Martens in the negative Schwarzian unimodal case and by Vargas in the case of C 2 multimodal maps without inflection points; see [14], [23] and also Shen's paper [18]. If all branches of f are monotone and there is at most one critical point (of inflection type), then such bounds were proved by Levin; see [8] and also [10].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 59%

Real bounds, ergodicity and negative Schwarzian for multimodal maps

Strien

¹

,

Vargas²

2004

J. Amer. Math. Soc.

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We consider smooth multimodal maps which have finitely many non-flat critical points. We prove the existence of real bounds. From this we obtain a new proof for the non-existence of wandering intervals, derive extremely useful improved Koebe principles, show that high iterates have ‘negative Schwarzian derivative’ and give results on ergodic properties of the map. One of the main complications in the proofs is that we allow f f to have inflection points.

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“…The following result was proved by Martens [16] in the case that f has negative Schwarzian, and extended to general smooth unimodal maps in [20], [11].…”

Section: Martens' Real Boundsmentioning

confidence: 88%