LIMITING DYNAMICS OF THE COMPLEX STANDARD FAMILY
Nuria Fagella
Dep. de Matematica Aplicada i Analisi
Universitat de Barcelona
Gran Via 585
08007 Barcelona
Spain
e-mail: fagella@maia.ub.es
ABSTRACT
The complexification of the standard family of circle maps
${\bf F}_{\alpha \beta}(\theta)=\theta+\alpha+\beta\sin(\theta) \bmod(2\pi)$
is given by
$F_{\alpha \beta}(\omega)=\omega e^{i \alpha} e^{(\beta/2)(\omega-1/ \omega)}$
and its lift $f_{\alpha \beta}(z)=z+\alpha+\beta \sin(z)$.
We investigate the $3$-dimensional parameter space for $F_{\alpha \beta}$
that results from considering $\alpha$ complex and $\beta$ real.
In particular, we study the $2$-dimensional cross sections $\beta=$constant
as $\beta$ tends to 0. As the functions tend to the rigid rotation
$F_{\alpha,0}$, their dynamics tend to the dynamics of the family
$G_\lambda(z)=\lambda z e^z$ where $\lambda=e^{-i \alpha}$.
This new family exhibits behavior typical of the exponential family
together with characteristic features of quadratic polynomials.
For example, we show that that the $\lambda$-plane contains infinitely
many curves for which the Julia set of the corresponding maps is the
whole plane. We also prove the existence of infinitely many sets of
$\lambda$ values homeomorphic to the Mandelbrot set.
REFERENCE
International Journal of Bifurcation and Chaos
Vol 3 (1995)
673-700