Abstract:
On the one hand, even the simplest nonlinear polynomials such as $f(z)=z^2+c$
demonstrate subtle and interesting features under iteration. On the other
hand, qualitative and quantitative characteristics discovered for the simplest
systems reappear in many more complicated systems (universality). Examples:
The Feigenbaum-Coullet-Tresser universality of bifurcation doubling cascades,
appearance of the Mandelbrot set in parameter planes of general analytic maps
depending holomorphically on the parameter (e.g., baby Mandelbrot sets in the
Mandelbrot set itself). An explanation of these phenomena is contained in the
concept of renormalization.
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