Bifurcations in the Space of Exponential Maps

Details Bifurcations in the Space of Exponential Maps Bifurcations in the Space of Exponential Maps

2003-11-26

arxiv.org/abs/math/0311480v6

Invent. Math. 175 (2009), No. 1, 103 - 135

Mathematics

Dynamical Systems

29 pages, 3 figures. The main change in the new version is the introduction of Theorem 1.1 on the connectivity of the bifurcat

Scientific paper

10.1007/s00222-008-0147-5

This article investigates the parameter space of the exponential family $z\mapsto \exp(z)+\kappa$. We prove that the boundary (in $\C$) of every hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as well as Baker and Rippon. In fact, we prove the stronger statement that the exponential bifurcation locus is connected in $\C$, which is an analog of Douady and Hubbard's celebrated theorem that the Mandelbrot set is connected. We show furthermore that $\infty$ is not accessible through any nonhyperbolic ("queer") stable component. The main part of the argument consists of demonstrating a general "Squeezing Lemma", which controls the structure of parameter space near infinity. We also prove a second conjecture of Eremenko and Lyubich concerning bifurcation trees of hyperbolic components.

Affiliated with

Also associated with

No associations

Rating

Bifurcations in the Space of Exponential Maps does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Bifurcations in the Space of Exponential Maps, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Bifurcations in the Space of Exponential Maps will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-122576