Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws
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Communications in
Mathematical Physics
Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws Oleg Kozlovski, Sebastian van Strien Imperial College, London, UK. E-mail: [email protected] Received: 12 July 2019 / Accepted: 19 May 2020 © The Author(s) 2020
Abstract: We consider a family of strongly-asymmetric unimodal maps { f t }t∈[0,1] of the form f t = t · f where f : [0, 1] → [0, 1] is unimodal, f (0) = f (1) = 0, f (c) = 1 is of the form and 1 − K − |x − c| + o(|x − c|) for x < c, f (x) = 1 − K + |x − c|β + o(|x − c|β ) for x > c, where we assume that β > 1. We show that such a family contains a Feigenbaum– Coullet–Tresser 2∞ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the 2∞ map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum–Coullet–Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results. Contents 1. 2. 3. 4. 5. 6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Setting of This Paper . . . . . . . . . . . . . . . . . . . . . . . . . . Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Background Material . . . . . . . . . . . . . . . . . . . . . . . . . Unusual Bifurcations of Families of Maps with Strong Asymmetries . . . The Existence of a 2∞ Map Within the Space of One-Sided Linear Unimodal Maps and a Full Family Result . . . . . . . . . . . . . . . . . . . . . . . 7. The Smallest Interval Argument . . . . . . . . . . . . . . . . . . . . . . . 8. Big Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Scaling Laws, Renormalization Limits and Universality . . . . . . . . . . 10. The Hausdorff Dimension of the Attracting Cantor Set is Zero . . . . . . .
O. Kozlovski, S. van Strien
11. Absence of any Koebe Space for General First Entry Maps . . . . . . . . . 12. Absence of Wandering Intervals . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction The theory of one-dimensional dynamics is rather well developed. Especially a lot is known for smooth one-dimensional unimodal maps (i.e. maps of an interval having just one critical point): absence of wandering intervals, real bounds, convergence of renormalizations, density of hyperbolic maps, various scaling properties, etc... Most of these results are obtained under some conditions on the order of the critical point, typically the map is assumed to be smooth or even analytic, and the critical point is assumed to be non-flat and in many results the order is, additionally, assumed to be an even integer. Moreover, in these studies, the order of the critical point is assumed to be the same on both sides, i.e. in a small neighbourho
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