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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Cycle doubling, merging, and renormalization in the tangent family
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by Tao Chen, Yunping Jiang and Linda Keen PDF
Conform. Geom. Dyn. 22 (2018), 271-314 Request permission


In this paper we study the transition to chaos for the restriction to the real and imaginary axes of the tangent family $\{ T_t(z)=i t\tan z\}_{0< t\leq \pi }$. Because tangent maps have no critical points but have an essential singularity at infinity and two symmetric asymptotic values, there are new phenomena: as $t$ increases we find single instances of “period quadrupling”, “period splitting”, and standard “period doubling”; there follows a general pattern of “period merging” where two attracting cycles of period $2^n$ “merge” into one attracting cycle of period $2^{n+1}$, and “cycle doubling” where an attracting cycle of period $2^{n+1}$ “becomes” two attracting cycles of the same period.

We use renormalization to prove the existence of these bifurcation parameters. The uniqueness of the cycle doubling and cycle merging parameters is quite subtle and requires a new approach. To prove the cycle doubling and merging parameters are, indeed, unique, we apply the concept of “holomorphic motions” to our context.

In addition, we prove that there is an “infinitely renormalizable” tangent map $T_{t_\infty }$. It has no attracting or parabolic cycles. Instead, it has a strange attractor contained in the real and imaginary axes which is forward invariant and minimal under $T^2_{t_\infty }$. The intersection of this strange attractor with the real line consists of two binary Cantor sets and the intersection with the imaginary line is totally disconnected, perfect, and unbounded.

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Additional Information
  • Tao Chen
  • Affiliation: Department of Mathematics, Engineering and Computer Science, Laguardia Community College, CUNY, 31-10 Thomson Avenue, Long Island City, New York 11101
  • MR Author ID: 1004078
  • Email:
  • Yunping Jiang
  • Affiliation: Department of Mathematics, Queens College of CUNY, Flushing, New York 11367; and Department of Mathematics, CUNY Graduate School, New York, New York 10016
  • MR Author ID: 238389
  • Email:
  • Linda Keen
  • Affiliation: Department of Mathematics, CUNY Graduate School, New York, New York 10016
  • MR Author ID: 99725
  • Email:
  • Received by editor(s): January 23, 2018
  • Received by editor(s) in revised form: July 26, 2018
  • Published electronically: November 28, 2018
  • Additional Notes: All three authors were partially supported by awards from PSC-CUNY. The second author was partially supported by grants from the Simons Foundation [grant number 523341], the NSF [grant number DMS-1747905), and the NSFC [grant number 11571122].
  • © Copyright 2018 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 22 (2018), 271-314
  • MSC (2010): Primary 37F30, 37F20, 37F10; Secondary 30F30, 30D30, 32A20
  • DOI:
  • MathSciNet review: 3880593