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Cycle doubling, merging, and renormalization in the tangent family. (English) Zbl 1407.37070
Summary: In this paper we study the transition to chaos for the restriction to the real and imaginary axes of the tangent family \(\{T_t(z)=it\tan z\}_{0<t\leq\pi}\). 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.

37F30 Quasiconformal methods and Teichm├╝ller theory, etc. (dynamical systems) (MSC2010)
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
30F30 Differentials on Riemann surfaces
30D30 Meromorphic functions of one complex variable (general theory)
32A20 Meromorphic functions of several complex variables
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