Tableaus and their use in Holomorphic Dynamics

MINICOURSE Tableaus and their use in Holomorphic Dynamics. TITLE of class I: Puzzles and para-puzzles, and the divergence property. ABSTRACT: The geometrical part: Puzzles in the most fundamental cases, i.e. associated with a polynomial, which either belongs to the Yoccoz-class of quadratic polynomials or to the bounded/unbounded-class of cubic polynomials (with one bounded and one un-bounded critical orbit).

Para-puzzles in the quadratic case. The analytical part: The divergence property and its usefulness. - An infinite set of open disjoint annuli embedded in a bounded open annuli, all of the same homotopy type, is said to have the divergence property, if the infinite series of moduli of these annuli is divergent.

TITLE of class II: Tableaus. ABSTRACT: The combinatorial part: Tableaus associated with a polynomial belonging to the Yoccoz-class or the bounded/unbounded class. The tableau rules, classification of critical tableaus, the Fibonacci critical tableau. TITLE of class III: Points are points. ABSTRACT: Combining the geometrical, analytical and combinatorial parts to conclude either local connectivity of the Julia set of a polynomial in the Yoccoz-class or total disconnectivity of the Julia set of a polynomial in the bounded/unbounded class (i.e. the Julia set is a Cantor set).

In both cases one proves that (certain) connected components are reduced to point components. Therefore, Adrien Douady liked to say that one proves that "points are points".