Cycle Doubling, Merging And Renormalization in the Tangent Family

TL;DR

This paper investigates the transition to chaos in tangent family maps, revealing novel phenomena like cycle merging and doubling, and employs renormalization and holomorphic motions to establish the existence and uniqueness of bifurcation parameters, along with identifying an infinitely renormalizable map with a strange attractor.

Contribution

It introduces new bifurcation phenomena in tangent maps, proves the existence and uniqueness of cycle doubling and merging parameters, and constructs an infinitely renormalizable map with a complex attractor.

Findings

Discovery of cycle merging and doubling phenomena.

Existence of a unique set of bifurcation parameters.

Identification of an infinitely renormalizable tangent map with a strange attractor.

Abstract

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.