# A priori bounds for bounded-primitive renormalization

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Jeremy Kahn, SUNY at Stony Brook
Fine Hall 401

We say that an infinitely renormalizable quadratic polynomial has bounded-primitive type if we can find an infinite sequence of primitive renormalization times, such that the ratio between consecutive terms of the sequence is bounded. We prove that any such polynomial has the a priori bounds: there is a lower bound on the modulus of all renormalizations. This implies that the Mandelbrot set is locally connected at the associated parameter values.