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