MLC at Feigenbaum points

We prove a priori bounds for Feigenbaum quadratic polynomials, i.e., infinitely renormalizable polynomials <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>z</mml:mi> <mml:mo>↦</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>c</mml:mi> </mml:mrow> </mml:math> of bounded type. It implies local connectivity of the corresponding Julia sets <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and MLC (local connectivity of the Mandelbrot set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℳ</mml:mi> </mml:math> ) at the corresponding parameters <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> . It also yields the scaling Universality, dynamical and parameter, for the corresponding combinatorics. The MLC Conjecture was open for the most classical period-doubling Feigenbaum parameter as well as for the complex tripling renormalizations. Universality for the latter was conjectured by Goldberg–Khanin–Sinai in the early 1980s.