Prerequisites: Familiarity with the complex plane, analysis in R^n

1. Summary of the basics of precise complex analysis (basic properties of holomorphic functions, Schwartz’s lemma, Arzelá-Ascoli theorem,…).
2. Linearization around periodic points (attractor/repulsor, super-attractor, parabolic and irrational cases.
3.Fatou and Julia sets, basic properties.
4. The quadratic family and the Mandelbrot set.
5. Almost conformal Cirigia and polynomial maps.
6. Copies of the Mandelbrot set on the Mandelbrot set (optional).

References:
BEARDON, A. – Iteration of Rational Functions. Complex analytic dynamical systems. Graduate Texts in Mathematics, 132. Springer-Verlag, New York, 1991.
CARLESON, L., T.Gamelin. – Complex Dynamics. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993.
de FARIA, E., de MELO, W. – Mathematical Tools for One-dimensional Dynamics – Cambridge Studies in Advanced Mathematics 115, Cambridge Universitry Press, (2008).
HUBBARD, J. H. – Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 2. Surface homeomorphisms and rational functions. Matrix Editions, Ithaca, NY, 2016
MILNOR, J. – Dynamics in one complex variable. Third edition. Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, 2006.
Mc MULLEN, C. T. – Complex Dynamics and renormalization – Annals of Mathemathics Studies 135, Princeton University Press (1994).