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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Renormalization towers and their forcing


Authors: Alexander Blokh and Michał Misiurewicz
Journal: Trans. Amer. Math. Soc. 372 (2019), 8933-8953
MSC (2010): Primary 37E15; Secondary 37E05, 37E20
DOI: https://doi.org/10.1090/tran/7928
Published electronically: August 20, 2019
MathSciNet review: 4029717
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Abstract: A cyclic permutation $ \pi :\{1, \dots , N\}\to \{1, \dots , N\}$ has a block structure if there is a partition of $ \{1, \dots , N\}$ into $ k\notin \{1,N\}$ segments of consecutive integers (blocks) of the same length, permuted by $ \pi $; call $ k$ the period of this block structure. Let $ p_1<\dots <p_s$ be periods of all possible block structures on $ \pi $. Call the finite string $ (p_1/1,$ $ p_2/p_1,$ $ \dots ,$ $ p_s/p_{s-1}, N/p_s)$ the renormalization tower of $ \pi $. The same terminology can be used for patterns, i.e., for families of cycles of interval maps inducing the same, up to the flip of the entire orbit, cyclic permutation (thus, there are two permutations, one of whom is a flip of the other one, that define a pattern). A renormalization tower $ \mathcal M$ forces a renormalization tower $ \mathcal N$ if every continuous interval map with a cycle of pattern with renormalization tower $ \mathcal M$ must have a cycle of pattern with renormalization tower $ \mathcal N$. We completely characterize the forcing relation among renormalization towers. Take the following order among natural numbers: $ 4\gg 6\gg 3\gg 8\gg 10\gg 5\gg \dots \gg 4n\gg 4n+2\gg 2n+1\gg \dots \gg 2\gg 1 $ understood in the strict sense (we write consecutive even numbers, starting with 4, then insert $ m$ after each number of the form $ 2(2s+1)$, and finally append the order with 2 and 1). We show that the forcing relation among renormalization towers is given by the lexicographic extension of this order. Moreover, for any tail $ T$ of this order there exists an interval map for which the set of renormalization towers of its cycles equals $ T$.


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Additional Information

Alexander Blokh
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294
Email: ablokh@math.uab.edu

Michał Misiurewicz
Affiliation: Department of Mathematical Sciences, Indiana University- Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202
Email: mmisiure@math.iupui.edu

DOI: https://doi.org/10.1090/tran/7928
Keywords: Sharkovsky order, forcing relation, cyclic patterns
Received by editor(s): November 24, 2018
Received by editor(s) in revised form: January 12, 2019, June 2, 2019, and June 5, 2019
Published electronically: August 20, 2019
Additional Notes: Research of the second author was supported by grant number 426602 from the Simons Foundation.
Article copyright: © Copyright 2019 American Mathematical Society