Last time, we reduced the proof of the exponential mixing property for expanding semiflows to the following Dolgopyat-like estimate:
Proposition 1 Let be an uniformly expanding Markov map on and let be a good roof function with exponential tails.Then, there exist , , and such that the iterates of the weighted transfer operator satisfy
for all and with , .
Remark 1 We use the same terminology from the previous post of this series.
Roughly speaking, the basic idea behind the exponential contraction property in Proposition 1 is that “oscillations produce cancellations”. In particular, the analysis of the “size” of is divided into two regimes:
- (A) If exhibits a high oscillation at scale (in the sense that ), then we will have a “cancelation” (significant reduction of the size of ) thanks to classical methods (Lasota-Yorke inequality);
- (B) If the oscillation of at scale is not high, then we will have a “cancelation” thanks to Dolgopyat’s mechanism, i.e., a combination of high oscillations of Birkhoff sums of the roof function (coming from the fact that is not a -coboundary) and the big phases , , of the terms in the formula defining .
In the remainder of this post, we will formalize this outline of proof of Proposition 1. More precisely, the next section contains a discussion of Lasota-Yorke inequality and the regime (A), and the last section is devoted to Dolgopyat’s cancelation mechanism and the regime (B).
1. Lasota-Yorke inequality and regime (A)
Before attacking Proposition 1, let us warm up with a digression on the spectral properties of the weighted transfer operators .
The usual transfer operator acts on the space of functions. This action has a simple isolated eigenvalue at . The eigenfunction associated to the eigenvalue and normalized so that is the density of the unique absolutely continuous invariant probability measure of . Furthermore, the essential spectral radius of is and has no eigenvalues of modulus except for . (See Aaronson’s book for more explanations.)
For close to , the operator is a small perturbation of . In particular, has an eigenfunction associated to its unique eigenvalue close to such that and converges to in the -topology as .
From now on, let us fix such that is well-defined and bounded away from zero for . (Here, is the constant appearing in the exponential tails condition for the roof function .)
The normalized weighted transfer operator satisfy and . In other words, if we replace by , we normalize both the eigenvalue and the eigenfunction to . Moreover, the proof of Proposition 1 can reduced to the analogous statement for the normalized operators:
for all and with and . Then, the conclusion of Proposition 1 is valid.
The proof of this proposition is based on Lasota-Yorke inequality (cf. Lemmas 7.8 and 7.9 of AGY paper):
- for all ;
- if , then ;
- if , then ,
where is the expansion constant of .
Before proving this lemma, let us use it to show Proposition 2. Recall that the assumption of this proposition is that
for some fixed , , and for all and with , , and our task is to prove the analogous statement in Proposition 1 for .
The idea is very simple: from the spectral discussion above, it is not hard to see that we introduce a factor of the order of when replacing by ; since is close to , this factor does not significantly affect definite contraction on the size of provided by the hypothesis of Proposition 2. Let us now turn into the details.
Given , we write with . Since and is uniformly close to , the previous estimate for the normalized operator gives that
for all with , . By the last item of Lemma 3, it follows that
Because is close to for small, we can choose such that
we deduce from the previous estimate that
1.1. Proof of Lasota-Yorke inequality
For the sake of convenience, we provide just a sketch of proof. We write
where is the set of inverse branches of , and (with ). By taking the derivative , we obtain five terms , , , , depending by differentiating , , , or .
The terms and are easy to deal with: the uniform bounds on and , and the contraction of inverse branches of imply that and . Thus, .
Similarly, the distortion bound (Renyi condition) on (see the previous post) implies that , so that .
Since and (because is a good roof function and the inverse branches of contract exponentially), we see that and, a fortiori, .
Finally, the exponential contraction of inverse branches of says that , so that . This proves the first item of Lemma 3.
The second item is an immediate consequence of the estimate and the first item just proved. Indeed, if , then
Closing the proof of Lemma 3, we observe that the third item is a direct corollary of the second item.
1.2. First part of Proposition 1: study of regime (A)
As we already mentioned, the first step in this direction is the treatment of the regime (A) of functions displaying high oscillations via Lasota-Yorke inequality:
Lemma 4 There exists such that any has the following property.Let with and and suppose that exhibits a high oscillation at scale in the sense that (where is the constant from Lemma 3). Then,
Proof: The facts that is normalized and has high oscillations imply that
Furthermore, the Lasota-Yorke inequality in the first item of Lemma 3 says that
Since and , we get that
for and sufficiently large (so that ).
Of course, we can iterate this lemma to establish the contraction property in Proposition 2 while the high oscillation property is not destroyed by :
Corollary 5 Fix where is the integer provided by Lemma 4. Suppose that is a function on and is an integer such that the high oscillation property is not destroyed by the first iterates of , i.e.,
for all and for some with , . Then,
Proof: By the third item of Lemma 3, we have that
By iterating Lemma 4, it follows that
This proves the corollary.
Remark 2 The proof of the corollary shows that a strong (pointwise/) form of cancellation occurs in the high oscillation case (regime (A)). As we are going to see in the next section, one has a much weaker () form of cancellation in the low oscillation case (regime (B)).
2. Dolgopyat’s mechanism and regime (B)
After our success in dealing with the regime (A) (cf. Corollary 5), let us analyze the regime (B) of functions whose oscillation is not high.
2.1. UNI condition
As we told in the introduction, the cancellation mechanism in regime (B) originates from the oscillations of the Birkhoff sums of the roof function . Here, by “oscillations of Birkhoff sums” we mean the following:
(where is the set of inverse branches of .)Then, the following conditions are equivalent:
- is not a coboundary: it is not possible to write with which is constant on each and .
- Uniform non-integrability (UNI) condition (“oscillation of Birkhoff sums”): there exists a constant such that
for some arbitrarily large and some inverse branches of .
This proposition is (contained in) Proposition 7.4 of AGY paper and we refer to it for the one page proof of this result.
Remark 3 The main point of this proposition is that a qualitative property (“not a coboundary”) in our definition of good roof function in the previous post turns out to be equivalent to a quantitative property (“definite oscillation of Birkhoff sums”). The nomenclature “uniform non-integrability” (UNI) comes from the fact that this is a (uniform) quantitative property issued from the fact that the suspension flow associated to and is not integrable (conjugated to a suspension flow with piecewise constant roof function) when is not a coboundary.
2.2. Dolgopyat’s cancellation mechanism
Let us now use UNI condition to produce non-trivial cancellations in some regions of the phase space for functions with “low oscillations”. In other terms, we want to study , , for such that .
For technical reasons, we will keep track of and an appropriate -bound for :
Definition 7 Given , we say that if is , is a function providing an adequate bound for in the sense that
- , and
for all .
Remark 4 is a cone and we are going to show in Lemma 10 that sends “strictly inside itself” in a certain sense.
Moreover, we need to anticipate the fact in Remark 3 that the cancellations in regime (B) occur only at certain spots of the phase space. In particular, the following “localization tool” (cf. Lemma 7.12 in AGY paper) will be helpful in the subsequent cancellation discussion.
This lemma is a consequence of the Lasota-Yorke inequality in Lemma 3 and its half-page proof can be found after the statement of Lemma 7.12 in AGY paper.
Remark 5 We call this lemma a “localization tool” for the following reason. We think of as a bump function supported on disjoint intervals of size . Next, we use to “localize” an appropriate bound in a pair to the support of by considering . In this setting, the lemma says that if the iterate of this localization gives a bound to , then is actually an appropriate bound for (i.e., ).
From now on, we use UNI condition (cf. Proposition 6) to fix and inverse branches such that
In this language, Dolgopyat’s cancellation mechanism can be stated as:
Lemma 9 There exist a small constant and a large constant with the following property. Let with , , and .Then, for every interval , we can find a point such that one of the following possibilities holds:
- Cancellation of type : for all , we have
- Cancellation of type : for all , we have
In other words, this lemma says that, for each interval of of size of order , we can find a subinterval of size of the same order where the two terms of
associated to the inverse branches and fixed above exhibits a significant cancellation w.r.t. the trivial bound (i.e., one gets a factor of instead of ).
This lemma is exactly Lemma 7.13 in AGY paper and we provide a sketch of its proof in the sequel.
Proof: Let , resp. be a small, resp. large constant. Consider and . Our task is to find with so that the conclusion of the lemma is valid.
The argument is divided into two cases. In the first (easy) case, we assume that the appropriate bound for is not “tight”, i.e., there exists such that or . In this situation, we want to take advantage of the non-tightness (and the fact that does not have high oscillations [as ]) to get some cancellation in an interval of size centered at .
For this sake, up to exchanging the roles of and , we can assume that . Since , we know that . Thus, because is a contraction. By Gronwall’s inequality, it follows that . Therefore,
for all . By integrating this estimate, we see that
for all . Since (by “non-tightness” assumption), we obtain from the estimates above that, if is small enough, then
for all . This proves the lemma in this “non-tight” case.
In the second case, we assume that the bound for is “tight”, i.e., and for all . In this context, we want to find such that the complex numbers
have opposite phases.
For this sake, we put for and we denote by the difference of the phases (arguments) of the complex numbers
Using UNI condition (2), the fact that does not have high oscillations (i.e., ) and the “tightness” of the bound for , it is possible to prove with a short calculation that
where . (See page 192 of AGY paper.)
It follows that if is sufficiently large (e.g., ), then there exists such that and have opposite phases.
We set . By exchanging the roles of and if necessary, we can assume that . By exploiting the fact that does not have high oscillations and the tightness of the bound for (via Gronwall inequality), it is possible to show (through another short calculation, cf. page 193 in AGY paper) that if is sufficiently small, then for all one has
where and denote the phases of the complex numbers and . In other terms, for any in the interval ,
- the size of does not drop too much in comparison with the size of : except for a factor , one still has the same comparison from the case ;
- and have almost opposite phases.
On the other hand, an elementary trigonometry lemma (cf. Lemma 7.14 in AGY paper) says that two complex numbers and such that
verify the “cancellation estimate”
By applying this lemma to and for , we deduce that if , then
that is, one has a cancellation of type .
This completes the sketch of proof of Lemma 9.
2.3. Second part of Proposition 1: study of regime (B)
Evidently, the key tool to get this -contraction property is Dolgopyat’s cancellation lemma 9 because it says that, given any interval of size , a non-trivial amount of cancellation for must happen inside a subinterval of comparable size . In particular, for the regime (B), even though we do not get cancellation everywhere in phase space, we still have cancellation in large chunks of the phase space. Thus, it is reasonable to expect a contraction property for in the norm (but not for the norm) in the regime (B).
Let us now try to formalize this heuristic argument. Fix such that where and are the constants in Lemma 9.
In other words, this lemma says that for any in regime (B) (i.e., ), we can find an appropriate bound for whose -norm got contracted by a definite factor in comparison with the -norm of an appropriate -bound for .
This lemma is exactly Lemma 7.15 in AGY paper. For the sake of convenience, let us now sketch its proof.
Proof: We take a maximal set of points such that the intervals are pairwise disjoint and compactly contained in . Note that the intervals cover .
By Dolgopyat’s cancellation lemma 9, we know that each contains a subinterval such that a cancellation of type or occurs for the pair in regime (B). We say that has type , resp. type depending on the type of cancellation occurring in .
We want to use our knowledge of the cancellation on each to modify the trivial bound for : roughly speaking, we want to insert a factor in front of the terms of associated to or whenever . At this point, the localization tool in Lemma 8 will prove itself useful.
From the technical point of view, we implement the idea from the previous paragraph as follows. We consider a bump function localized on each : we impose on , outside and (for some universal constant ). Next, we construct on by putting
Note that for some constant depending only on the fixed objects , and . In particular, is independent of and the pair . The function was built up so that is a function taking values in such that and
has a definite contraction factor in front of the term associated to or whenever for some .
In summary, we manipulated some bump functions associated to to get a function such that:
- , ,
- for and given by the type of ,
- for or and not of the type of .
From the properties in the last two items, we see that the statement of Dolgopyat’s cancellation lemma 9 implies that
that is, is a bound for . By combining this fact with the property in the first item and the localization tool in Lemma 8, we have that is actually an appropriate bound for , i.e.,
Therefore, our task is reduced to prove that for some constant (independent of , and ). In this direction, a short (half-page) calculation (cf. page 193 in AGY paper) exploiting the cancellation mechanism for any shows that
Note that the estimate (3) goes in the good direction: it implies that
by the definition of the normalized transfer operator (and the fact that ), and we know that for sufficiently small.
On the other hand, the estimate (4) by itself is not sufficient to control in the desired way.
Fortunately, we know that covers “most” of the phase space and does not oscillate too much: more precisely, we have that is formed of intervals of size contained in some intervals covering the whole phase space (as it was said in the beginning of the proof) and .
By combining this facts with Gronwall’s inequality, it is possible to prove (after a short calculation, cf. pages 195 and 196 of AGY paper) that there exists a constant such that
Because the intervals are pairwise disjoint, the intervals cover , the density is bounded away from and , and is uniformly bounded (since the size of both and is ), the estimate above implies that there exists a constant such that
This new information (coming from the low oscillations of and the fact that covers almost all of ) permits to get the desired -contraction for . Indeed, this is a consequence of the following simple computation (resembling to “Peter-Paul inequality”). Denote by . By (3) and (4), we have
for any parameter . By applying (5) to the right-hand side of the previous inequality, we deduce that
for each . By taking so that , we see that the estimate above implies that
Therefore, if we take small enough so that , then
This completes the proof of the lemma.
Corollary 11 Consider the integer fixed above. Let , and be the constants provided by Lemma 10, and denote by with , .Let and . Suppose that is the first time such that does not exhibit high oscillation at scale , i.e.,
Proof: We set . By definition, where is the constant function . By successively applying Lemma 10, we obtain a sequence of pairs such that
By setting and by recalling that for all , we conclude from the previous estimate that
This proves the corollary.