Variations of the Pommerenke–Levin–Yoccoz inequality

The Pommerenke–Levin–Yoccoz (PLY) inequality describes the dynamics of a complex (single variable) polynomial near a repelling fixed point, namely, it relates the repelling strength with the combinatorial rotation number. A well-known consequence of the PLY inequality is the local connectivity of the Mandelbrot set at points of the main cardioid. I will talk about a generalization of the PLY inequality that is applicable not only to fixed points but also to certain repelling invariant continua. This is a joint project with A. Blokh, G. Levin, and L. Oversteegen.