Last time, we introduced the notion of *regular parameter* of the quadratic family and we saw that the orbits of have a nice statistical description when is regular. In particular, this reduced our initial goal (of proving Jakobson’s theorem) to show that regular parameters are abundant near , i.e.,

As it turns out, Yoccoz’s proof of (1) is indirect: first, he introduces the notion of *strongly regular parameter* and he proves that strongly regular parameters are a special case of regular parameters; secondly, he exploits the nice features of strong regularity to *transfer* some key properties from the phase space to the parameter space in order to prove that

Today, we shall define strong regularity and prove the regularity of such parameters (while leaving the proof of (2) for the final post of this series).

**1. Some preliminaries **

** 1.1. Quick review of the regularity property **

For , has two fixed points and with . Note that the critical value belongs to .

In a certain sense, the key idea is to study the dynamics of via certain intervals (“Yoccoz puzzle pieces”) bounded by points in the pre-orbits of .

For example, the notion of regular parameter was defined with the aid of the intervals and where is given by . Indeed, is *regular* if there are and such that

for all . Here, is called –*regular* if there are and an interval such that sends diffeomorphically onto in such a way that . For later use, we denote by the inverse branch of restricted to .

In general, any -regular point belongs to a *regular interval* of order , that is, an interval possessing an open neighborhood such that sends diffeomorphically onto in such a way that . In other words, the set of -regular points is the union of regular intervals of orders .

It is easy to describe regular intervals (“Yoccoz puzzle pieces”) in terms of the pre-orbits of . In fact, denote by (so that and ). It is not difficult to check that if is a regular interval of order and is the associated neighborhood, then are *consecutive* points of and are *consecutive* points of .

** 1.2. Dynamically meaningful partition of the parameter space **

For later use, we organize the parameter space as follows. For each , we consider a maximal open interval such that is the first return of to under for all .

In analytical terms, we can describe the sequence as follows. For , let be and, for , define recursively as

In these terms, is the solution of the equation .

Remark 1From this analytical definition of , one can show inductively that for along the following lines.By definition, . This inductive relation can be exploited to give that for all and .

This estimate allows us to show that the function has derivative between and for . Since this function takes a negative value at and a positive value at , we see that this function has a unique simple zero such that for , as desired.

Remark 2Note that is a decreasing sequence such that for some universal constant . Indeed, the function takes the value at (cf. Subsection 4.2 of the previous post), it vanishes at , and it has derivative between and , so that .

From now on, we think of where is a *large* integer.

**2. Strong regularity **

Given , let be the collection of maximal regular intervals of positive order contained in and consider

the function for , and the map ( for ): cf. Subsection 4.3.3 of this post here.

Remark 3Even though is not contained in any element of , we set and for .

The elements of of “small” orders are not hard to determine. Given , define by:

It is not difficult to check that the *sole* elements of of order are the intervals

and, furthermore, any other element of has order .

The intervals , , are called *simple regular intervals*: this terminology reflects the fact that they are the most “basic” type of regular intervals.

In this setting, a parameter is *strongly regular* if “most” of the returns of to occur on simple regular intervals:

Definition 1We say that is strongly regular up to level if and, for each , one has

A parameter is called strongly regular if it is strongly regular of all levels .

Remark 4Let be a strongly regular parameter. It takes a while before encounters a non-simple regular interval: if (or, equivalently, ), then (3) implies that

where . In particular, , so that the first iterates of encounter exclusively at simple regular intervals.

**3. Regularity of strongly regular parameters **

Let us now outline the proof of the fact that strongly regular parameters are regular.

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