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:
Definition 1 We 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 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
.
Remark 5 For
and
, there is only one
-singular interval, namely
. For
and
, there are exactly three
-singular intervals, namely
and
.
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
:
Proposition 4 Let
be a strongly regular parameter up to level
. Then,
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.
Corollary 5 If
is strongly regular up to level
and
, then
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
).
Corollary 7 Let
be strongly regular up to level
and fix
. Then, the total measure of peripheral
-singular intervals is
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.
Corollary 10 Assume that
is a stationary level. Then,
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.
By exploiting this structure result in a similar way to the arguments in Corollaries 7 and 10, one can show (with a half-page long proof) the following estimate:
Corollary 12 Assume that
is a non-stationary level. Then,
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 definition of central, lateral and peripheral intervals,
By Corollary 5, the first term satisfies:
The right hand side is at most whenever
i.e.,
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
By induction hypothesis, we conclude that
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:
Thus,
for large enough.
The last two inequalities together imply that
Finally, by plugging the estimates (5), (6) and (7) into (4), we deduce that
for large enough. This proves the desired theorem.
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