It is well-known that the local connectivity of the Mandelbrot set reduces to proving suitable bounds for the renormalizations for infinitely renormalizable quadratic polynomials. I will discuss joint work with M. Lyubich where we prove these bounds for renormalizations coming from the "elephant eyes" of the Mandelbrot set. It is natural to prove these bounds first for the geometric limit of the elephant eyes. I will review at the beginning the concepts of geometric limits and renormalization.
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