On some properties of an $\exp(iz)$ map

I. V. Matyushkin

Russian journal of nonlinear dynamics, Tome 12 (2016) no. 1, pp. 3-15.

Résumé

The properties of an $e^{iz}$ map are studied. It is proved that the map has one stable and an infinite number of unstable equilibrium positions. There are an infinite number of repellent twoperiodic cycles. The nonexistence of wandering points is heuristically shown by using MATLAB. The definition of helicity points is given. As for other hyperbolic maps, Cantor bouquets are visualized for the Julia and Mandelbrot sets.

Détail

Keywords: holomorphic dynamics, hyperbolic map.
Mots-clés : fractal, Cantor bouquet

Bibliographie

[1] Milnor J. W., Dynamics in one complex variable: Introductory lectures, Ann. of Math. Stud., 160, 3rd ed., Princeton Univ. Press, Princeton, 2006, 320 pp. | MR

[2] Mandelbrot B. B., Fractals: Form, chance, and dimension, Freeman, San Francisco, Calif., 1977, 365 pp. | MR | Zbl

[3] Alexander D., Devaney R. L., “A century of complex dynamics”, A century of advancing mathematics, eds. S. F. Kennedy, D. J. Albers, G. L. Alexanderson, D. Dumbaugh, F. A. Farris, D. B. Haunspe, Math. Assoc. America, Washington, D.C., 2015, 15–34 | MR

[4] Devaney R. L., “${e}^{z}$: Dynamics and bifurcation”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1:2 (1991), 287–308 | DOI | MR | Zbl

[5] Devaney R. L., “Cantor bouquets, explosions, and Knaster continua: Dynamics of complex exponentials”, Publ. Mat., 43:1 (1999), 27–54 | DOI | MR | Zbl

[6] Kapoor G. P., Guru Prem Prasad M., “Chaotic burst in the dynamics of a class of noncritically finite entire functions”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9:6 (1999), 1137–1151 | DOI | MR | Zbl

[7] Kapoor G. P., Guru Prem Prasad M., “Dynamics of $(e\sp z-1)/z$: The Julia set and bifurcation”, Ergodic Theory Dynam. Systems, 18:6 (1998), 1363–1383 | DOI | MR | Zbl

[8] Guru Prem Prasad M., “Chaotic burst in the dynamics of $f\sb \lambda(z)=\lambda\frac{{\rm sinh}(z)}{z}$”, Regul. Chaotic Dyn., 10:1 (2005), 71–80 | DOI | MR | Zbl

[9] Moreno Rocha M., On indecomposable subsets of the Julia set for unstable exponentials, PhD dissertation, Boston Univ., Boston, 2002, 79 pp.

[10] Moreno Rocha M., “Existence of indecomposable continua for unstable exponentials”, Topology Proc., 27:1 (2003), 233–244 | MR | Zbl

[11] Kotus J., ““Cantor bouquets” for non-entire meromorphic functions”, Workshop on Cantor bouquets in hedgehogs and transcendental iteration (june 16th-19th 2009, Toulouse, France) http://www.math.univ-toulouse.fr/anr_abc/bouquet/Slides/Janina.pdf

[12] Mihaljević-Brandt H., Topological dynamics of transcendental entire functions, PhD Dissertation, Univ. of Liverpool, Liverpool, 2009, 137 pp. | Zbl

[13] Rempe-Gillen L., Sixsmith D., Hyperbolic entire functions and the Eremenko – Lyubich class: Class $\mathcal{B}$ or not class $\mathcal{B}$?, 2015, 20 pp., arXiv: 1502.00492 [math.CV]

[14] Al-Husseiny H. F., “A study of the dynamics of the family $\lambda \frac{\sinh ^{m} z}{z^{2m} }$”, Iraqi J. Sci., 52:4 (2011), 494–503

[15] Urbański M., Zdunik A., “Real analyticity of Hausdorff dimension of finer Julia sets of exponential family”, Ergodic Theory Dynam. Systems, 24:1 (2004), 279–315 | DOI | MR | Zbl

[16] Schleicher D., “Dynamics of entire functions”, Holomorphic dynamical systems, Lectures given at the {C.I.M.E.} Summer School (Cetraro, Italy, July 7–12, 2008), Lecture Notes in Math., 1998, eds. G. Gentili, J. Guenot, and G. Patrizio, Springer, Berlin, 2010, 295–339 | DOI | MR | Zbl

[17] Ghys É., Goldberg L. R., Sullivan D. P., “On the measurable dynamics of $z\mapsto e^z$”, Ergodic Theory Dynam. Systems, 5:3 (1985), 329–335 | DOI | MR | Zbl

[18] Aarts J. M., Oversteegen L. G., “The geometry of Julia sets”, Trans. Amer. Math. Soc., 338:2 (1993), 897–918 | DOI | MR | Zbl