×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Arneodo, A.; Coullet, P.; Tresser, C., A possible new mechanism for the onset of turbulence, Phys. Lett. A, 81, 4, 197-201 (1981)
[2] Tresser, Charles; Coullet, Pierre, It\'{e}rations d’endomorphismes et groupe de renormalisation, C. R. Acad. Sci. Paris S\'{e}r. A-B, 287, 7, A577-A580 (1978) · Zbl 0402.54046
[3] CK T. Chen and L. Keen, Dynamics of Generalized Nevanlinna Functions, arXiv:1805.10974 (2018).
[4] Devaney, Robert L.; Keen, Linda, Dynamics of tangent. Dynamical systems, College Park, MD, 1986-87, Lecture Notes in Math. 1342, 105-111 (1988), Springer, Berlin
[5] Douady, Adrien, Syst\`“emes dynamiques holomorphes. Bourbaki seminar, Vol. 1982/83, Ast\'”{e}risque 105, 39-63 (1983), Soc. Math. France, Paris
[6] Douady, A., Chirurgie sur les applications holomorphes. Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Berkeley, Calif., 1986, 724-738 (1987), Amer. Math. Soc., Providence, RI
[7] FK N. Fagella and L. Keen, Stable components in the parameter plane of meromorphic functions of finite type, arXiv:1702.06563 (2018).
[8] Feigenbaum, Mitchell J., Quantitative universality for a class of nonlinear transformations, J. Statist. Phys., 19, 1, 25-52 (1978) · Zbl 0509.58037
[9] Feigenbaum, Mitchell J., The universal metric properties of nonlinear transformations, J. Statist. Phys., 21, 6, 669-706 (1979) · Zbl 0515.58028
[10] Gardiner, Frederick P.; Jiang, Yunping; Wang, Zhe, Holomorphic motions and related topics. Geometry of Riemann surfaces, London Math. Soc. Lecture Note Ser. 368, 156-193 (2010), Cambridge Univ. Press, Cambridge · Zbl 1198.30019
[11] Jiang, Yunping, Geometry of Cantor systems, Trans. Amer. Math. Soc., 351, 5, 1975-1987 (1999) · Zbl 0916.58032
[12] Jiang, Yunping, Renormalization and geometry in one-dimensional and complex dynamics, Advanced Series in Nonlinear Dynamics 10, xvi+309 pp. (1996), World Scientific Publishing Co., Inc., River Edge, NJ · Zbl 0864.58018
[13] Keen, Linda, Complex and real dynamics for the family \(\lambda\tan({\bf z})\), S\={u}rikaisekikenky\={u}sho K\={o}ky\={u}roku, 1269, 93-102 (2002)
[14] Keen, Linda; Kotus, Janina, Dynamics of the family \(\lambda\tan z\), Conform. Geom. Dyn., 1, 28-57 (1997) · Zbl 0884.30019
[15] Keen, Linda; Kotus, Janina, On period doubling phenomena and Sharkovskii type ordering for the family \(\lambda\tan(z)\). Value distribution theory and complex dynamics, Hong Kong, 2000, Contemp. Math. 303, 51-78 (2002), Amer. Math. Soc., Providence, RI · Zbl 1020.30024
[16] LSS G. Levin, S. van Strien, and W. Shen, Monotonicity of entropy and positively oriented transversality for families of interval maps, arXiv:1611.10056v1 (2016).
[17] McMullen, Curtis T., Complex dynamics and renormalization, Annals of Mathematics Studies 135, x+214 pp. (1994), Princeton University Press, Princeton, NJ · Zbl 0822.30002
[18] de Melo, Welington; van Strien, Sebastian, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 25, xiv+605 pp. (1993), Springer-Verlag, Berlin · Zbl 0791.58003
[19] Milnor, John, Dynamics in one complex variable, viii+257 pp. (1999), Friedr. Vieweg & Sohn, Braunschweig · Zbl 0946.30013
[20] Milnor, John, On the concept of attractor, Comm. Math. Phys., 99, 2, 177-195 (1985) · Zbl 0595.58028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.