The idea of Yoccoz puzzles for non-renormalizable (or finitely renormalizable) quadratic polynomials is similar to the idea of Markov partitions in smooth dynamical systems. However, one difference is that there is a sequence of 2-to-1 pieces in a Yoccoz puzzle, which are called central pieces. So most interesting study of a Yoccoz puzzle is concentrated in the sequence of central pieces. Application of the Yoccoz puzzles are a proof of the local connectivity of the Julia set of a non-renormalizable (or finitely renormalizable) quadratic polynomial by Yoccoz, proofs of the local connectivity of the Mandelbrot set at non-renormalizable (or finitely renormalizable) points by Yoccoz and by Shishikura (the combining of the holomorphic motion in Shishikura's proof makes it more interesting), and a proof of the zero Lebesgue measure of the Julia set of a non-renormalizable (or finitely renormalizable) quadratic polynomial by Shishikura.
In this talk I will study three-dimensional puzzles for infinitely renormalizable quadratic polynomials. I will show how to use the three-dimensional puzzle for an infinitely renormalizable quadratic polynomial to construct a most natural sequence of simple renormalizations. I will prove the modulus inequality in renormalization [1]. One example of an application of the three-dimensional puzzles is the proof of quasisymmetric property of the conjugacy between two Feigenbaum-like one-dimensional maps [2]. Another application of the three-dimensional puzzles is a proof of the local connectivity of the Julia set of a certain infinitely renormalizable quadratic polynomial [1]. I will also show how to use the three-dimensional puzzles to study the local connectivity of the Mandelbrot set at certain infinitely renormalizable points near Misiurewicz points (thus the union of these points is dense on the boundary of the Mandelbrot set and the corresponding Julia set for each of these points is locally connected) [3]. I will also show a study of a conformal measure for an infinitely renormalizable quadratic polynomial by using the three-dimensional puzzle (an ongoing research with Huang and Wang) [4].