Almost twenty years ago, Jean-Christophe Yoccoz gave several lectures (at ETH and Collège de France, for instance) on a proof of Jakobson theorem (based on the so-called Yoccoz puzzles), and he distributed hard copies of his lecture notes whenever they were requested (for example, he gave me such a copy when I became his post-doctoral student in 2007).

More recently, he gave a couple of lectures at Collège de France aiming to generalize his proof of Jakobson theorem to obtain some results improving upon the Wang-Young theory of rank one attractors.

Of course, his lectures (whose videos are available at his website) started by recalling several elements of his proof of Jakobson theorem and I decided to take the opportunity to write a series of posts about Jean-Christophe’s proof of Jakobson theorem (based on his hand-written lecture notes from 1997).

For the first installment of this series, we’ll review some aspects of the dynamics of the one-dimensional quadratic maps and, after that, we’ll state Jakobson’s theorem, describe the three main steps of Yoccoz proof of Jakobson theorem and implement the first step of the strategy.

**1. Introduction **

The dynamics of one-dimensional affine maps is fairly easy understand. The change of variables provided by the translation transforms into . If , the choice shows that the affine map is conjugated to its linear part . If , is a translation.

In other words, the dynamics of polynomial maps is not very interesting when . On the other hand, we shall see below that quadratic maps are already sufficiently complicated to produce all kinds of rich dynamical behaviors.

**2. Quadratic family **

The *quadratic family* is where and .

Remark 1This family is sometimes presented in the literature as or . As it turns out, these presentations are equivalent to each other: indeed, the affine change of variables converts into

The dynamics of near infinity is easy to understand: in fact, since for , one has that attracts the orbit of any with .

This means that the interesting dynamics of occurs in the filled-in Julia set:

Note that is totally invariant, that is, . Also, is a compact set because implies that and whenever , so that

(where ).

Moreover, because it contains all periodic points of (i.e., all solutions of the algebraic equations , ).

Remark 2is a full compact set, i.e., is connected: indeed, this happens because the maximum principle implies that a bounded open set with boundary must be completely contained in (i.e., ).

The dynamics of on is influenced by the behaviour of the orbit of the critical point . More precisely, let us consider the Mandelbrot set

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