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 1 From 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 2 Note 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 3 Even 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:
Remark 4 Let 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.