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The Mandelbrot set and its copies

Abstract

The Mandelbrot set classifies the behaviour of the quadratic polynomials, andit is a central object in complex dynamics. A fascinating fact is the presence of copies of the Mandelbrot set in the Mandelbrot set itself and in many other parameter planes. Within the Mandelbrot set there exists two different kind of little copies: the primitive copies (with a cusp), and the satellite ones (without a cusp). It has been proven by Lyubich that primitive copies of the Mandelbrot set are quasiconformally homeomorphic to the Mandelbrot set itself. It has been general belief that the satellite copies of the Mandelbrot set are mutually quasiconformally homeomorphic. In a previous work joint with C. Petersen, we disprove this expectation, showing that in general the satellite copies of the Mandelbrot set are not quasiconformally homeomorphic.Anyway, we believe and we are working for proving that some satellite copies (with rotation number with the same denominator) are quasiconformally homeomorphic. On the other hand, the Mandelbrot set seems to appear in the parameter plane of a significant family of holomorphic correspondences, which are mating between quadratic maps and the modular group. This family has been introduced by S. Bullett and C. Penrose, who conjectured that the connectedness locus (this is, the set of parameters for which the limit set is connected)for this family is homeomorphic to the Mandelbrot set. Since then there has been partial results,but the conjecture is still open. Together with S. Bullett, we are working on giving a dynamical theory for this family of correspondences, and to solve the conjecture. (AU)