The differentiability of the hairs of 𝑒𝑥𝑝(𝑍)

da Silva, M. Viana

doi:10.1090/S0002-9939-1988-0955004-1

The differentiability of the hairs of $\textrm {exp}(Z)$
HTML articles powered by AMS MathViewer

by M. Viana da Silva PDF

Proc. Amer. Math. Soc. 103 (1988), 1179-1184 Request permission

Abstract:

We prove that the hairs of the exponential-like maps $f(z) = \lambda {e^z}$ are smooth curves. This answers affirmatively a question of Devaney and Krych. The proof is constructive in the sense that a dynamically defined ${C^\infty }$ parametrization is presented.

References

MichałMisiurewicz, On iterates of $e^{z}$, Ergodic Theory Dynam. Systems 1 (1981), no. 1, 103–106. MR 627790, DOI 10.1017/s014338570000119x
Robert L. Devaney and MichałKrych, Dynamics of $\textrm {exp}(z)$, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 35–52. MR 758892, DOI 10.1017/S014338570000225X
Robert L. Devaney, Exploding Julia sets, Chaotic dynamics and fractals (Atlanta, Ga., 1985) Notes Rep. Math. Sci. Engrg., vol. 2, Academic Press, Orlando, FL, 1986, pp. 141–154. MR 858012
Robert L. Devaney, Julia sets and bifurcation diagrams for exponential maps, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 167–171. MR 741732, DOI 10.1090/S0273-0979-1984-15253-4
Robert L. Devaney and Folkert Tangerman, Dynamics of entire functions near the essential singularity, Ergodic Theory Dynam. Systems 6 (1986), no. 4, 489–503. MR 873428, DOI 10.1017/S0143385700003655