The Attractor of Fibonacci-like Renormalization Operator
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
The Attractor of Fibonacci-like Renormalization Operator Hao Yang JI1)
Si Min LI
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P. R. China E-mail : [email protected] [email protected] Abstract In this paper we extend the Fibonacci-like maps to a wider class with the so-called “bounded combinatorics”. The Fibonacci-like renormalization operator R is defined and we show that the orbit of each map from this class converges to a universal limit under iterates of R. Keywords
Fibonacci combinatorics, principal nest, renormalization
MR(2010) Subject Classification
1
37E05, 37F25
Introduction
The Fibonacci combinatorics appeared in the context of unimodal interval maps related to Milnor’s attractor problem [19] about the classification of the measure-theoretical attractors in one-dimensional dynamics. Existence of real polynomials with Fibonacci combinatorics can be obtained by kneading theory, see [18]. For example, among the real polynomial family Pc (z) = z + c, there is a unique one whose critical point has this combinatorics. And as it implies the strong recurrence of the critical point, it was considered in [5] as a candidate for a map having a wild attractor. A wild attractor (also called absorbing Cantor attractor) for smooth unimodal map is a compact invariant Cantor set whose basin of attraction is a meager subset with full Lebesgue measure. It was proved by Lyubich and Milnor [15] that quadratic real Fibonacci polynomial has no wild attractor. This result was then generalized to any combinatorics by Shen [20] through a purely real argument. It was also observed in [15] that the geometry of Fibonacci maps for critical order = 2 and ≥ 4 were different. On the other hand, it was proved in [2] that real Fibonacci unimodal map with critical order sufficiently high may have a wild attractor. After that, in [1] Bruin generalized the results of [2] to a larger class of unimodal maps. For smooth unimodal maps with non-flat critical point this problem was reduced to the case that f is non-renormalizable with a non-periodic recurrent critical point. In order to study the geometry of the critical set ω(c) of a non-renormalizalbe map, Lyubich introduced the concept of generalized renormalization in [13]. This transforms the class of unimodal maps to a class of maps with a single critical point but defined on the union of disjoint intervals mapped onto a bigger interval. Any unimodal map with recurrent critical point including Fibonacci map Received May 5, 2019, revised October 11, 2019, accepted June 23, 2020 The second author is supported by NSFC (Grant No. 11731003) 1) Corresponding author
Fibonacci-like Renormalization
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becomes infinitely renormalizable in this sense. Following this point of view, in [12] Li and Wang described some combinatorial types which are generalized “Fibonacci-like” but fail to satisfy Bruin’s condition significantly. As the para
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