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.
3.1. Singular intervals
Given , we say that an interval is –singular if its boundary consists of two consecutive points of , but is not contained in a regular interval of order . The collection of -singular intervals is denoted by .
By definition, .
For later reference, denote . In these terms, is regular whenever there are such that
As a “warm-up”, let us show the following elementary fact:
for all and .
Proof: For , we have that is a singleton (cf. Remark 5). Moreover,
when (cf. Subsection 4.2 of the previous post). Since the function is increasing on , we get the desired estimate for .
On the other hand, if , then
This completes the proof of the proposition.
3.2. Central, peripheral and lateral intervals
The analysis of for requires the introduction of certain (combinatorially defined) neighborhoods of the critical point and the critical value .
Assume that for some . For each , let be the element such that .
Denote by the decreasing sequence of regular intervals containing the critical value defined recursively as follows: is a regular interval of order and is the regular interval of order determined by its inverse branch
Also, let us consider and . Here, if is a regular interval, then , where and , , , and in general.
Remark 6 By definition, the endpoints of are the points of immediately adjacent to the endpoints of .
Note that is the connected component of containing the critical point , while the endpoints of are adjacent in to the endpoints of . Here (and in the sequel), .
We say that an interval is central, lateral or peripheral depending on its relative position with respect to :
Definition 3 Let be a strongly regular parameter up to a level such that . An interval is called:
- central whenever ;
- lateral if but ;
- peripheral if .
3.3. Measure estimate for central intervals
We shall control the total measure of central intervals by estimating the Lebesgue measure of :
for all .
Proof: is the neighborhood of the critical point of the quadratic map defined by . Therefore, is comparable to :
By the usual distortion estimates (cf. Subsection 4.3 of the previous post), it is possible to check that is comparable to :
This reduces our task to estimate . Since the interval has a fixed size and for some (as ), it suffices to control for . For this sake, we recall that the derivative of is not far from a “coboundary”:
where (see Proposition 6 of the previous post for a motivation of in the case ). In particular,
By exploiting this estimate, one can show (with a one-page long argument) that the strong regularity up to level of implies a (strong form of) Collet-Eckmann condition:
for all and . Because for and for , the proof of the proposition is complete.
3.4. Measure estimates for peripheral intervals
We control the total measure of peripheral intervals by relating them to singular intervals of lower order. More concretely, a (half-page long) combinatorial argument provides the following structure result for the generation of peripheral intervals:
Proposition 6 (Structure of peripheral intervals) Let be strongly regular up to level and consider . If is a peripheral interval, then:
- either has the form for some ,
- or has the form for some ,
where (is a regular interval of order ).
Proof: A point in a peripheral interval is not close to : indeed, (by definition) and (by Corollary 5 for ). Hence,
The previous proposition says that if is a peripheral interval, then has the form with or with . From the fact that the derivative of is an “almost coboundary” (cf. the proof of Proposition 4 above), one can show that:
- when ;
- when .
Therefore, for some or and, a fortiori,
for some or .
It follows that
so that the proof of the corollary is complete.
3.5. Measure estimates for lateral intervals
The anlysis of lateral intervals is combinatorially more involved. For this reason, we subdivide the class of lateral intervals into stationary and non-stationary:
Definition 8 Let be strongly regular up to level , fix and consider a lateral interval. For each , either or (because ).The level of is the largest integer such that . (Note that and .
We say that the level is stationary if .
The strategy to control the total measure of lateral intervals is similar to argument used for peripheral intervals: we want to exploit structure results describing the construction of lateral intervals out of singular intervals of lower orders. As it turns out, the case of lateral intervals with stationary level is somewhat easier (from the combinatorial point of view) and, for this reason, we start by treating this case.
For later use, we denote
3.5.1 Lateral intervals with stationary levels
Denote by (a regular interval of order ) and . The structure of lateral intervals with stationary levels is given by the following proposition:
Proposition 9 Let be a lateral interval with stationary level . Suppose that reverses the orientation. Then, has the form for some contained in . A similar statement (with replacing ) holds when preserves orientation.
The proof of this proposition is short, but we omit it for the sake of discussing the total measure of lateral intervals with stationary level.
Proof: The argument is very close to the case of peripheral intervals (i.e., Corollary 7). In fact, if is a lateral interval with stationary level and reverses the orientation, then for some with . Note that implies that . Since , the usual bounded distortion properties say that
for any and, a fortiori,
This completes the proof of the corollary because (cf. the proof of Proposition 4).
3.5.2 Lateral intervals with non-stationary levels
The structure of lateral intervals with non-stationary levels is the following:
Proposition 11 Let be a lateral interval with non-stationary level . Suppose that reverses the orientation. Then,
- either for some ,
- or where is a regular interval of order (with right endpoint immediately to the left of in ) and .
A similar statament holds when preserves orientation.
Moreover, if , then one has a better estimate:
3.6. Proof of regularity of strongly regular parameters
The measure estimates developed in the last three subsections permit to establish the main result of this post, namely:
Theorem 13 Fix . If is large enough depending on (i.e., ), then for any strongly regular up to level and any we have
In particular, any strongly regular parameter is regular.
Proof: We will show this theorem by induction. The initial cases of this theorem were already established in Proposition 2.
Suppose that and for all . Replacing by a smaller integer (if necessary), we can assume that .
By Corollary 5, the first term satisfies:
The right hand side is at most whenever
We have two possibilities:
- if , then for large enough (e.g., );
- if , we have for large enough thanks to the strong regularity assumption (cf. Remark 4).
The contribution of peripheral intervals is controlled by induction hypothesis. More precisely, by Corollary 7, one has
The contribution of lateral intervals is estimated as follows. Fix . The bounds on in the case of a non-stationary level are worse than in the case of a stationary level : compare Corollaries 10 and 12. For this reason, we will use only the bounds coming from the non-stationary situation in the sequel.
If is not simple, i.e., its order is , then the induction hypothesis (applied to ) implies that
Since for large enough and (cf. Remark 4), we conclude that
for large enough.
If is simple (i.e., ), then we use the second part of Corollary 12 and the induction hypothesis to obtain:
for large enough.
for large enough. This proves the desired theorem.