Posted by: matheuscmss | May 17, 2015

Decay of correlations for flows and Dolgopyat’s estimate II.a

In a series of two posts, we will revisit our previous discussion on the exponential mixing property for hyperbolic flows via a technique called Dolgopyat’s estimate.

Here, our main goal is to provide a little bit more of details on how this technique works by offering a “guided tour” through Sections 2 and 7 of a paper of Avila-Goüezel-Yoccoz.

For this sake, we organize this short series of posts as follows. In next section, we introduce a prototypical class of semiflows exhibiting exponential mixing. After that, we state the main exponential mixing result of this post (an analog of Theorem 7.3 of Avila-Goüezel-Yoccoz paper) for such semiflows, and we reduce the proof of this mixing property to a Dolgopyat-like estimate on weighted transfer operators. Finally, the next post of the series will be entirely dedicated to sketch the proof of the Dolgopyat-like estimate.

1. Expanding semiflows

Recall that a suspension flow is a semiflow {T_t:\Delta_r\rightarrow \Delta_r}, {t\in\mathbb{R}_+}, associated to a base dynamics (discrete-time dynamical system) {T:\Delta\rightarrow\Delta} and a roof function {r:\Delta\rightarrow\mathbb{R}^+} in the following way. We consider {\Delta_r:=(\Delta\times\mathbb{R}^+)/\sim} where {\sim} is the equivalence relation induced by {(T(x),0)\sim (x,r(x))}, and we let {T_t} be the semiflow on {\Delta_r} induced by

\displaystyle (x,s)\in \Delta\times\mathbb{R}^+\mapsto (x,s+t)\in\Delta\times\mathbb{R}^+

Geometrically, {T_t}, {0\leq t<\infty} flows up the point {(x,s)}, {0\leq s<r(x)}, linearly (by translation) in the fiber {\{x\}\times\mathbb{R}^+} until it hits the “roof” (the graph of {r}) at the point {(x,r(x))}. At this moment, one is sent back (by the equivalence relation {\sim}) to the basis {\Delta\times\{0\}} at the point {(T(x),0)\sim (x,r(x))}, and the semiflow restarts again.

A more concise way of writing down {T_t} is the following: denoting by {\Delta_r:=\{(x,t):x\in \Delta, 0\leq t<r(x)\}}, one defines {T_t(x,s) := (T^n x, s+t-r^{(n)}(x))} where {r^{(n)}(x)} is the Birkhoff sum

\displaystyle r^{(n)}(x):=\sum\limits_{k=0}^{n-1} r(T^k x) \ \ \ \ \ (1)

and {n} is the unique integer such that

\displaystyle r^{(n)}(x)\leq s+t<r^{(n+1)}(x)

In this post, we want to study the decay of correlations of expanding semiflows, that is, a suspension flow {T_t} so that the base dynamics {T} is an uniformly expanding Markov map and the roof function {r} is a good roof function with exponential tails in the following sense.

Remark 1 Avila-Gouëzel-Yoccoz work in greater generality than the setting of this post: in fact, they allow {\Delta} to be a John domain and they prove results for excellent hyperbolic semiflows (which are more common in “nature”), but we will always take {\Delta=(0,1)} and we will study exclusively expanding semiflows (which are obtained from excellent hyperbolic semiflows by taking the “quotient along stable manifolds”) in order to simplify our exposition.

Definition 1 Let {\Delta=(0,1)}, {Leb} be the Lebesgue measure on {\Delta}, and {\{\Delta^{(l)}\}_{l\in L}} be a finite or countable partition of {\Delta} modulo zero into open subintervals. We say that {T:\bigcup\limits_{l\in L} \Delta^{(l)}\rightarrow \Delta} is an uniformly expanding Markov map if

  • {\{\Delta^{(l)}\}} is a Markov partition: for each {l\in L}, the restriction of {T} to {\Delta^{(l)}} is a {C^1}-diffeomorphism between {\Delta^{(l)}} and {\Delta};
  • {T} is expanding: there exist a constant {\kappa>1} and, for each {l\in L}, a constant {C(l)>1} such that {\kappa\leq|T'(x)|\leq C(l)} for each {x\in\Delta^{(l)}};
  • {T} has bounded distortion: denoting by {J(x)=1/|T'(x)|} the inverse of the Jacobian of {T} and by {\mathcal{H}=\{(T|_{\Delta^{(l)}})^{-1}\}_{l\in L}} the set of inverse branches of {T}, we require that {\log J} is a {C^1} function on each {\Delta^{(l)}} and there exists a constant {C>0} such that

    \displaystyle \left|\frac{h''(x)}{h'(x)}\right| = |D((\log J)\circ h)(x)|\leq C

    for all {h\in \mathcal{H}} and {x\in \Delta}. (This condition is also called Renyi condition in the literature.)

Remark 2 Araújo and Melbourne showed recently that, for the purposes of discussing exponential mixing properties (for excellent hyperbolic semiflows with one-dimensional unstable subbundles), the bounded distortion (Renyi condition) can be relaxed: indeed, it suffices to require that {\log J} is a Hölder function such that the Hölder constant of {\log J\circ h} is uniformly bounded for all {h\in \mathcal{H}}.

Example 1 Let {\Delta=\Delta^{(0)}\cup\Delta^{(1)}=(0,1/2)\cup(1/2,1)} be the finite partition (mod. {0}) of {\Delta} provided by the two subintervals {\Delta^{(l)}=(\frac{l}{2}, \frac{l+1}{2})}, {l=0, 1}. The map {T:\Delta^{(0)}\cup\Delta^{(1)}\rightarrow\Delta} given by {T(x)=2x-l} for {x\in\Delta^{(l)}} is an uniformly expanding Markov map (preserving the Lebesgue measure {Leb}).

An uniformly expanding map {T} preserves an unique probability measure {\mu} which is absolutely continuous with respect to the Lebesgue measure {Leb}. Moreover, the density {d\mu/dLeb} is a {C^1} function whose values are bounded away from {0} and {\infty}, and {\mu} is ergodic and mixing.

Indeed, the proof of these facts can be found in Aaronson’s book and it involves the study of the spectral properties of the so-called transfer (Ruelle-Perron-Frobenius) operator

\displaystyle Lu(x) = \sum\limits_{T(y)=x} J(y) u(y) = \sum\limits_{h\in\mathcal{H}} J(hx) u(hx)

(as it was discussed in Theorem 1 of Section 1 of this post in the case of a finite Markov partition {\{\Delta^{(l)}\}_{l\in L}}, {\# L<\infty})

Definition 2 Let {T:\bigcup\limits_{l\in L} \Delta^{(l)}\rightarrow \Delta} be an uniformly expanding Markov map. A function {r:\bigcup\limits_{l\in L}\Delta^{(l)}\rightarrow\mathbb{R}^+} is a good roof function if

  • there exists a constant {\varepsilon>0} such that {r(x)\geq\varepsilon} for all {x};
  • there exists a constant {C>0} such that {|D(r\circ h)(x)|\leq C} for all {x} and all inverse branch {h\in\mathcal{H}} of {T};
  • {r} is not a {C^1}coboundary: it is not possible to write {r = \psi + \phi\circ T - \phi} where {\psi:\Delta\rightarrow\mathbb{R}} is constant on each {\Delta^{(l)}} and {\phi:\Delta\rightarrow\mathbb{R}} is {C^1}.

Remark 3 Intuitively, the condition that {r} is not a coboundary says that it is not possible to change variables to make the roof function into a piecewise constant function. Here, the main point is that we have to avoid suspension flows with piecewise constant roof functions (possibly after conjugation) in order to have a chance to obtain nice mixing properties (see this post for more comments).

Definition 3 A good roof function {r:\bigcup\limits_{l\in L}\Delta^{(l)}\rightarrow\mathbb{R}^+} has exponential tails if there exists {\sigma_0 > 0} such that {\int_{\Delta} e^{\sigma_0 r} d Leb < \infty}.

The suspension flow {T_t} associated to an uniformly expanding Markov map {T:\bigcup\limits_{l\in L}\Delta^{(l)}\rightarrow\Delta} and a good roof function {r:\bigcup\limits_{l\in L}\Delta^{(l)}\rightarrow\mathbb{R}^+} with exponential tails preserves the probability measure

\displaystyle \mu_r:=\mu\otimes Leb/\mu\otimes Leb(\Delta_r)

on {\Delta_r}. Note that {\mu_r} is absolutely continuous with respect to {Leb_r:= Leb\otimes Leb} (because {\mu} is absolutely continuous with respect to {Leb}).

Remark 4 All integrals in this post are always taken with respect to {Leb} or {Leb_r} unless otherwise specified.

Remark 5 In the sequel, AGY stands for Avila-Gouëzel-Yoccoz.

2. Statement of the exponential mixing result

Let {(T_t)_{t\in\mathbb{R}}} be an expanding semiflow.

Theorem 4 There exist constants {C>0}, {\delta>0} such that

\displaystyle \left|\int U\cdot V\circ T_t \, d Leb_r - \left(\int U \, d Leb_r\right) \left(\int V\,d\mu_r\right)\right|\leq C e^{-\delta t}\|U\|_{C^1}\|V\|_{C^1}

for all {t\geq 0} and for all {U, V\in C^1(\Delta_r)}.

Remark 6 By applying this theorem with {U(x,t)\cdot \frac{d\mu}{d Leb}(x)} in the place of {U}, we obtain the classical exponential mixing statement:

\displaystyle \left|\int U\cdot V\circ T_t \, d\mu_r - \left(\int U \, d \mu_r\right) \left(\int V\,d\mu_r\right)\right|\leq C e^{-\delta t}\|U\|_{C^1}\|V\|_{C^1}

Remark 7 This theorem is exactly Theorem 7.3 in AGY paper except that they work with observables {U} and {V} belonging to Banach spaces {\mathcal{B}_0} and {\mathcal{B}_1} which are slightly more general than {C^1} (in the sense that {C^1\subset \mathcal{B}_0\subset \mathcal{B}_1}). In fact, AGY need to deal with these Banach spaces because they use their Theorem 7.3 to deduce a more general result of exponential mixing for excellent hyperbolic semiflows (see their paper for more explanations), but we will not discuss this point here.

The remainder of this post is dedicated to the proof of Theorem 4.

2.1. Reduction of Theorem 4 to Paley-Wiener theorem

From now on, we fix two observables {U,V\in C^1} such that

\displaystyle \int V \, d\mu_r=0 \ \ \ \ \ (2)

Of course, there is no loss in generality here because we can always replace {V} by {V-\int V d\mu_r} if necessary.

In this setting, we want to show that the correlation function

\displaystyle C(t):=\int U\cdot V\circ T_t \, d Leb_r \ \ \ \ \ (3)

decays exponentially.

For this sake, we will use the following classical theorem in Harmonic Analysis (stated as Theorem 7.23 in AGY paper):

Theorem 5 (Paley-Wiener) Let {\rho:\mathbb{R}_+\rightarrow\mathbb{R}} be a bounded measurable function and denote by {\widehat{\rho}(s):=\int_{x=0}^{\infty} e^{-sx}\rho(x) dx} (defined for {s\in\mathbb{C}} with {\textrm{Re}(s)>0}) the Laplace transform of {\rho}.Suppose that {\widehat{\rho}} can be analytically extended to a function {\phi} defined on a strip {\{s=\sigma+it\in\mathbb{C}: |\sigma|<\varepsilon, t\in\mathbb{R}\}} in such a way that

\displaystyle \int_{t=-\infty}^{\infty}\sup\limits_{|\sigma|<\varepsilon} |\phi(\sigma+it)| dt < \infty

Then, there exists a constant {C>0} and a full measure subset {A\subset\mathbb{R}} such that

\displaystyle |\rho(x)|\leq C e^{-\varepsilon x/2}

for all {x\in A}.

In other words, the Paley-Wiener theorem says that a bounded measurable function decays exponentially whenever its Laplace transform admits a nice analytic extension to a vertical strip to the left of the imaginary axis {\textrm{Re}(s)=0}.

In our context, we will produce such an analytic extension by writing the Laplace transform of the correlation function as an appropriate geometric series of terms depending on {s}. In fact, the {k}th term of this series will have a clear dynamical meaning: it will be related to the pieces of orbit {\{T_s(x): 0\leq s\leq t\}} hitting {k} times the graph of the roof function.

More concretely, for each {t\geq 0}, we decompose the phase space as

\displaystyle \Delta_r := A_t\cup B_t

where

  • {A_t=\{(x,a)\in\Delta_r: a+t\geq r(x)\}} is the subset of points {(x,a)\in\Delta_r} whose piece of orbit {\{T_s(x,a):0\leq s\leq t\}} hits the graph of the roof function at least once,
  • and {B_t:=\Delta_r-A_t} is the subset of points {(x,a)\in\Delta_r} whose piece of orbit {\{T_s(x,a):0\leq s\leq t\}} does not hit the graph of the roof function.

Let us denote by

\displaystyle C(t) = \int_{A_t} U\cdot V\circ T_t \, dLeb_r + \int_{B_t} U\cdot V\circ T_t \, dLeb_r := \rho(t) + (C(t)-\rho(t))

the corresponding decomposition of the correlation function (3).

The fact that the roof function {r} has exponential tails implies that the probability of the event that the piece of orbit {\{T_s(x,a): 0\leq s\leq t\}} does not hit the graph of {r} becomes exponentially small for {t} large.

Thus, it is not surprising that the following calculation shows that {C(t)-\rho(t)} decays exponentially fast as {t} grows:

\displaystyle \begin{array}{rcl} |C(t)-\rho(t)| &\leq & \|U\|_{C^0} \|V\|_{C^0}\int_{x\in\Delta}\max\{r(x)-t,0\} \\ &\leq & \|U\|_{C^0} \|V\|_{C^0} \int\limits_{\{x\in\Delta: r(x)\geq t\}} r(x) \\ &\leq & \|U\|_{C^0} \|V\|_{C^0} \|r\|_{L^2} \sqrt{Leb(\{x\in\Delta: r(x)\geq t\})} \\ &\leq & C\|U\|_{C^0}\|V\|_{C^0} e^{-\sigma_0 t/2} \end{array}

where {C=\|r\|_{L^2}\int e^{\sigma_0r}<\infty} (by the exponential tails condition on {r}).

In particular, the proof of Theorem 4 is reduced to show that {\rho(t)} decays exponentially. As we already mentioned, the basic idea to achieve this goal is to use the Paley-Wiener theorem and, for this reason, we want to write the Laplace transform as an appropriate series by decomposing the phase space accordingly to the number {k\geq 1} of times that a certain piece of orbit hits the graph of the roof function:

\displaystyle \begin{array}{rcl} \widehat{\rho}(s) &=& \int_{x\in\Delta} \int_{a=0}^{r(x)} \int_{t+a\geq r(x)} e^{-st}\cdot U(x,a) \cdot V\circ T_t(x,a) \, dt \, da \, d Leb(x) \\ &=& \sum\limits_{k=1}^{\infty} \int_{x\in\Delta} \int_{a=0}^{r(x)} \int_{b=0}^{r(T^k x)} U(x,a) \cdot V(T^k x,b) e^{-s(b+r^{(k)}(x)-a)}\, db \, da \, d Leb(x) \end{array}

where {r^{(k)}(x)} is defined by (1). Note that this is valid for any {s\in\mathbb{C}} with {\textrm{Re}(s)>0}.

An economical way of writing down this series uses the partial Laplace transform of a function {W:\Delta_r\rightarrow\mathbb{R}} defined by the formula {\widehat{W}(s):=\int_0^{r(x)} W(x,a) e^{-sa} \, da} for {s\in\mathbb{C}}. In this language, the previous identity gives that

\displaystyle \widehat{\rho}(s) = \sum\limits_{k=1}^{\infty} \int_{x\in\Delta} \widehat{U}_{-s}(x) \cdot e^{-sr^{(k)}(x)} \cdot \widehat{V}_s(T^k x) \, d Leb(x)

By a change of variables {T^k(x) = y} (i.e., “duality”), we obtain that

\displaystyle \widehat{\rho}(s) = \sum\limits_{k=1}^{\infty}\int_{\Delta} \left(L_s^k\widehat{U}_{-s}\right)(y)\cdot \widehat{V}_{s}(y) dLeb(y), \quad \forall\, s\in\mathbb{C}, \, \textrm{Re}(s)>0, \ \ \ \ \ (4)

where {L_s^k} is the {k}th iterate of the weighted transfer operator

\displaystyle L_su(y):=\sum\limits_{T(z)=y} e^{-sr(z)} J(z) u(z) = \sum\limits_{h\in\mathcal{H}} e^{-s r(hy)} J(hy) u(hy) \ \ \ \ \ (5)

Remark 8 In general, we expect the terms of the series (4) to decay exponentially because {T} is expanding and {L_s} is “dual” to the (Koopman-von Neumann) operator defined by composing functions with {T}.

This formula (stated as Lemma 7.17 in AGY paper) is the starting point for the application of Paley-Wiener theorem to {\widehat{\rho}}. More precisely, we can exploit it to analytically extend {\widehat{\rho}} into three steps:

  • (a) Lemma 7.21 in AGY paper: {\widehat{\rho}(s)} can be extended to a neighborhood of any {s=it\neq 0};
  • (b) Lemma 7.22 in AGY paper: {\widehat{\rho}(s)} can be extended to a neighborhood of the origin {s=0};
  • (c) Corollary 7.20 in AGY paper: {\widehat{\rho}(s)} can be extended to a strip {\{s=\sigma+it\in\mathbb{C}: |\sigma|\leq\sigma_1, |t|\geq T_0\}} for some {\sigma_1>0} small and {T_0>0} large in such a way that the extension {\phi} satisfies {|\phi(s)|\leq C/t^2} (for some constant {C>0} depending on {U} and {V}).

Geometrically, the first two steps permit to extend {\widehat{\rho}(s)} to rectangles of the form

\displaystyle \{s=\sigma+it: |\sigma|\leq \sigma_1(T), |t|\leq T\}

for {T>0}. Indeed, this is a consequence of (a), (b) and the compactness of the segment {\{s=it: |t|\leq T\}}.

Of course, these steps are not sufficient to extend {\widehat{\rho}} to a whole strip to the left of the imaginary axis (because there is nothing preventing {\sigma_1(T)\rightarrow 0} as {T\rightarrow\infty}), and this is why the third step is crucial.

In summary, the combination of the three steps (a), (b) and (c) (and the fact that the function {1/t^2} is integrable) says that {\widehat{\rho}} has an analytic extension {\phi} to a strip {\{\sigma+it: |\sigma|<\sigma_2, t\in\mathbb{R}\}} satisfying the hypothesis of Paley-Wiener theorem 5.

Therefore, we have that {\rho(t)} decays exponentially (and, a fortiori, the correlation function {C(t)} also decays exponentially) if we can establish (a), (b) and (c).

2.2. Implementation of (a), (b) and part of (c)

Observe that the series in (4) is bounded by

\displaystyle \left|\sum\limits_{k=1}^{\infty}\int_{\Delta} \left(L_s^k\widehat{U}_{-s}\right)(y)\cdot \widehat{V}_{s}(y) dLeb(y)\right| \leq \sum\limits_{k=1}^{\infty} \|\widehat{V}_s\|_{L^2} \cdot \|L_s^k\widehat{U}_{-s}\|_{L^2} \ \ \ \ \ (6)

This hints that we should first compare the sizes of {\widehat{V}_s} and {\widehat{U}_{-s}} to the sizes of {V} and {U} before trying to show the geometric convergence of this series (for certain values of {s}). In this direction, we have the following pointwise bound (compare with the equation (7.66) in AGY paper):

Lemma 6 There exists a constant {C>0} such that

\displaystyle |\widehat{W}_s(x)|\leq \frac{C\|W\|_{C^1}}{\max\{1,|t|\}}e^{\sigma_0 r(x)/2}

for any {C^1} function {W:\Delta_r\rightarrow\mathbb{R}} and for all {s=\sigma+it} with {|\sigma|\leq \sigma_0/4}.

Proof: Recall that {\widehat{W}_s(x):=\int_0^{r(x)} W(x,a) e^{-sa}\,da}. Hence, the desired estimate is trivial for {|t|\leq 1}. The remaining case {|t>1} is dealt with by integration by parts. Indeed, we have

\displaystyle \widehat{W}_s(x) = \left[W(x,a)\frac{e^{sa}}{s}\right]_{0}^{r(x)} - \int_0^{r(x)} \partial_a W(x,a) \frac{e^{sa}}{s} \,da

Since the right-hand side has boundary terms bounded by {C\|W\|_{C_0}e^{\sigma_0 r(x)/4}/|t|} and an integral term bounded by

\displaystyle C\|W\|_{C^1} r(x)e^{\sigma_0 r(x)/4}/|t|\leq C' \|W\|_{C^1} e^{\sigma_0 r(x)/2}/|t|

(as {r(x)\geq\varepsilon_1>0}), we see that the proof of the lemma is complete. \Box

This pointwise bound permits to control {\|\widehat{V}_s\|_{L^2}} (cf. Lemma 7.18 in AGY paper):

Corollary 7 There exists a constant {C>0} such that

\displaystyle \|\widehat{V}_s\|_{L^2}\leq \frac{C\|V\|_{C^1}}{\max\{1,|t|\}}

for all {s=\sigma+it} with {|\sigma|\leq \sigma_0/4}.

Proof: This is an immediate consequence of Lemma 6 and the fact that the function {e^{\sigma_0 r}} is integrable (by the exponential tails condition on {r}). \Box

On the other hand, this pointwise bound is not adequate to control {\|L_s^k\widehat{U}_{-s}\|_{L^2}} in terms of a geometric series. In fact, it is well-known that the weighted transfer operators only exhibit some contraction property when one also works with stronger norms than {L^2}. In our current setting, it might be tempting to try to control the {C^1} norm of {\widehat{U}_{-s}} in terms of the {C^1} norm of {U}. As we are going to see now, this does not quite work for {\widehat{U}_{-s}} directly (as the pointwise bound in Lemma 6 also involves the function {e^{\sigma_0 r(x)/2}} which might be unbounded), but it does work for {L_s\widehat{U}_{-s}} (compare with Lemma 7.18 of AGY paper):

Lemma 8 There exists a constant {C>0} such that

  • {\|L_s\widehat{U}_{-s}\|_{C^0}\leq C\|U\|_{C^1}/\max\{1,|t|\}};
  • {\|DL_s\widehat{U}_{-s}\|_{C^0}\leq C\|U\|_{C^1}},

for all {s=\sigma+it} with {|\sigma|\leq \sigma_0/4}.

Proof: Recall that {L_s\widehat{U}_{-s}(x) = \sum\limits_{h\in\mathcal{H}} e^{-sr(hx)} J(hx) \widehat{U}_{-s}(hx)}.

It follows from Lemma 6 that

\displaystyle \begin{array}{rcl} |L_s\widehat{U}_{-s}(x)| &\leq& C\frac{\|U\|_{C^1}}{\max\{1,|t|\}}\sum\limits_{h\in\mathcal{H}} e^{\sigma_0 r(hx)/4} J(hx) e^{\sigma_0 r(hx)/2} \\ &\leq& C\frac{\|U\|_{C^1}}{\max\{1,|t|\}}L_{3\sigma_0/4}(1) \end{array}

where {1} denotes the constant function of value one. Since it is not hard to check that the operator {L_{\sigma}} acting on the space of {C^1} functions is bounded for {\sigma<\sigma_0}, the first item of the lemma is proved.

Let us now prove the second item of the lemma. For this sake, we write {L_s\widehat{U}_{-s}(x)= \sum\limits_{h\in\mathcal{H}} e^{-sr(hx)} J(hx) \int_0^{r(hx)}U(hx,a) e^{sa}\,da}. In particular, the derivative {DL_s\widehat{U}_{-s}(x)} has four terms: one can differentiate the term {e^{sr(hx)}} or {J(hx)} or the limit of integration or {U(hx,a)} (resp.). Let us denote by {(I)}, {(II)}, {(III)} and {(IV)} (resp.) the terms obtained in this way.

The fourth term is bounded by

\displaystyle \begin{array}{rcl} |(IV)|&\leq& C\|DU\|_{C^0}\sum\limits_{h\in\mathcal{H}} e^{\sigma_0 r(hx)/4} J(hx) r(hx)e^{\sigma_0 r(hx)/4} \\ &\leq& C\|DU\|_{C^0} L_{3\sigma_0/4}(1) \leq C\|U\|_{C^1} \end{array}

Similarly, the bounded distortion property {\|D(J\circ h)(x)\|\leq C J(hx)} for {T} implies that

\displaystyle |(II)| \leq C\|U\|_{C^0}\sum\limits_{h\in\mathcal{H}} e^{\sigma_0 r(hx)/4} J(hx) r(hx)e^{\sigma_0 r(hx)/4}\leq C\|U\|_{C^0}

Finally, since {\|D(r\circ h)\|_{C^0}\leq C} (by definition of good roof function) and {D(e^{-sr(hx)}) = -s e^{-sr(hx)}}, we see that the third term is bounded by

\displaystyle |(III)|\leq C\|U\|_{C^1}

and the first term is bounded by

\displaystyle |(I)|\leq C|s|\cdot |L_s\widehat{U}_{-s}(x)|\leq C\frac{|s|}{\max\{1,|t|\}}\|U\|_{C^1}

(thanks to the estimate from the first item.)

This completes the proof of the lemma. \Box

Remark 9 An important point in this lemma is that the {C^0} norm of {L_s\widehat{U}_{-s}} behaves differently from the {C^0} norm of {DL_s\widehat{U}_{-s}}.This suggests that we should measure {C^1} functions using the norm

\displaystyle \|u\|_{1,t}:=\|u\|_{C^0} + \frac{1}{\max\{1,|t|\}}\|Du\|_{C^0} \ \ \ \ \ (7)

in order to get some uniform control on the operator {L_s}: indeed, this norm allows to rewrite the previous lemma as

\displaystyle \|L_s\widehat{U}_{-s}\|_{1,t}\leq \frac{C\|U\|_{C^1}}{\max\{1,|t|\}}

and this is exactly the statement of Lemma 7.18 in AGY paper. This norm will show up again in the statement of the Dolgopyat-like estimate.

Once we have Corollary 7 and Lemma 8, we are ready use the estimate (6) and some classical properties (namely, Lasota-Yorke inequality and weak mixing for {T_t}) in order to implement the step (a) of the “Paley-Wiener strategy”:

Lemma 9 For any {s=it\neq 0}, there exists an open disk {O_s} (of radius independent of {U} and {V}) centered at {s} such that {\widehat{\rho}(s)} has an analytic extension to {O_s}.

Proof: It is well-known (cf. Lemma 7.8 in AGY paper) that the weighted transfer operator {L_{it}} acting on {C^1} satisfies a Lasota-Yorke inequality. We will come back to this point in the next post of this series, but for now let us just mention a key spectral consequence of a Lasota-Yorke inequality for {L_{it}}.

By Hennion’s theorem (cf. Baladi’s book), a Lasota-Yorke inequality for {L_{it}} implies that its essential spectral radius is {<1} and its spectral radius {\leq 1}. In concrete terms, this means that there exists a constant {0<\gamma<1} such that the spectrum of {L_{it}} is entirely contained in the ball {\{z\in\mathbb{C}: |z|<\gamma\}} except for possibly finitely many eigenvalues (counted with multiplicity) located in the annulus {\{z\in\mathbb{C}: \gamma\leq |z|\leq 1\}}.

In other words, if one can show that {L_{it}} has no eigenvalues of modulus {1}, then the previous description of the spectrum of {L_{it}} gives that {\|L_{it}^n\|_{C^1}\leq C r^n} for some constants {C>0} and {r<1}, and for all {n\in\mathbb{N}}. Of course, since {L_s} is a small analytic perturbation of {L_{it}} for any {s} in a small disk {O_{it}} centered at {it}, this implies that

\displaystyle \|L_s^{n}\|_{C^1}\leq C_{\ast} r_{\ast}^n

for some constants {C_{\ast}>0}, {r_{\ast}<1} and for all {s\in O_{it}}, {n\in\mathbb{N}}. In particular, by combining this estimate with Corollary 7, Lemma 8 and (6), we obtain that in this setting the series

\displaystyle \phi(s):=\sum\limits_{k=1}^{\infty} \int_{\Delta} \widehat{V}_s\cdot L_s^{k-1}(L_s\widehat{U}_{-s})

defines an analytic extension to {O_{it}} of {\widehat{\rho}(s)}.

In summary, we have reduced the proof of the lemma to the verification of the fact that {L_{it}} has no eigenvalues of modulus {1} (when {it\neq0}). As it turns out, this is an easy consequence of the spectral characterization of the weak-mixing property for the expanding semiflow {T_t}: indeed, this property says that the Koopman-von Neumann operator given by composition with {T_t} has no eigenvalues {\lambda\neq 1} of modulus {|\lambda|=1}, and it is not difficult to see that this means that {L_{it}} has no eigenvalues of modulus {1}.

This proves the lemma. \Box

Remark 10 A more direct proof of this lemma (without relying on the weak-mixing property for {T_t}) can be found in Lemma 7.21 of AGY paper.

Next, let us adapt the argument above to perform the step (b) of the Paley-Wiener strategy:

Lemma 10 There exists an open disk {O_0} (of radius independent of {U} and {V}) centered at {0} such that {\widehat{\rho}(s)} has an analytic extension to {O_0}.

Proof: The transfer operator {L_0} has a simple eigenvalue {1} (cf. Aaronson’s book). In particular, the argument used to prove the previous lemma does not work (i.e., it is simply false that {\|L_0^k\|_{C^1}} decays exponentially as {k\rightarrow\infty}).

Nevertheless, we can overcome this difficulty as follows. For {s} in a small open disk {O_0} centered at {0}, {L_s} is an analytic perturbation of {L_0}. Thus, {L_s} has an eigenvalue {\lambda_s} close to {1}, and we can write

\displaystyle L_s := \lambda_s P_s + R_s

where {P_s(f)} is the spectral projection to the eigenspace generated by the normalized eigenfunction {f_s} (with {\int f_s=1}) associated to {\lambda_s}, and {P_sR_s=R_sP_s=0}. Furthermore, the spectral properties of {L_s} mentionned during the proof of Lemma 9 also tell us that there exist uniform constants {C>0} and {r<1} such that {\|R_s^n\|_{C^1}\leq Cr^n} for all {s\in O_0} and {n\in\mathbb{N}}.

In other terms, after we remove from {L_s} the component associated to the eigenvalue {\lambda_s}, we obtain an operator {R_s} with nice contraction properties.

At this point, the basic idea is to “repeat” the argument of the proof of Lemma 9 with {L_s} replaced by {R_s}. In this direction, we rewrite the series (4) as

\displaystyle \begin{array}{rcl} \widehat{\rho}(s) &=& \sum\limits_{n=1}^{\infty} \int \widehat{V}_s\cdot L_s^n\widehat{U}_{-s} = \sum\limits_{k=0}^{\infty} \int \widehat{V}_s\cdot L_s^k(L_s\widehat{U}_{-s}) \\ &=& \frac{1}{1-\lambda_s}\int \widehat{V}_s\cdot P_s(L_s\widehat{U}_{-s}) + \sum\limits_{k=0}^{\infty} \int \widehat{V}_s\cdot R_s^k(L_s\widehat{U}_{-s}) \end{array}

(Here, we used that {L_s=\lambda_sP_s+R_s}, {P_sR_s=R_sP_s=0}, and {\frac{1}{1-\lambda_s} = \sum\limits_{k=0}^{\infty}\lambda_s^k})

Observe that the series {\sum\limits_{k=0}^{\infty} \int \widehat{V}_s\cdot R_s^k(L_s\widehat{U}_{-s})} converges for all {s\in O_0} thanks to Corollary 7, Lemma 8 and the fact that {\|R_s^k\|_{C^1}\leq Cr^k} with {r<1} for all {s\in O_0}.

It follows that we can use the previous equation to define an analytic extension of {\widehat{\rho}} to {O_0} if we can control the term {\frac{1}{1-\lambda_s}\int\widehat{V}_s\cdot P_s(L_s\widehat{U}_{-s})}. In other words, the proof of the lemma is reduced to show that

\displaystyle O_0\ni s\mapsto \frac{1}{1-\lambda_s}\int\widehat{V}_s\cdot P_s(L_s\widehat{U}_{-s}) \ \ \ \ \ (8)

is an analytic function.

Note that this is a completely obvious task for {s\in O_0} because {\lambda_0=1}, i.e., the analytic function

\displaystyle O_0\ni s\mapsto \frac{1}{1-\lambda_s}

has a pole at {s=0}.

Fortunately, the order of pole at {s=0} of this function can be shown to be one by the following calculation. Since {L_s} is an analytic perturbation of {L_0}, we have that {\|L_s-L_0\|_{C^1}=O(s)} and {\|f_s-f_0\|_{C^1}=O(s)}. In particular,

\displaystyle \begin{array}{rcl} \lambda_s &=& \int L_sf_s = \int (L_s-L_0)(f_s-f_0) + \int L_0 (f_s-f_0) + \int L_s f_0 \\ &=& O(s^2) + \int(f_s-f_0) + \int L_0(e^{-sr}f_0) = O(s^2) + 0 + \int e^{-sr}d\mu \\ &=& 1 - s \int rd\mu + O(s^2) \end{array}

where {\mu=f_0\cdot Leb} is the absolutely continuous invariant probability measure of {T}. This means that {s\mapsto\lambda_s} has derivative {\int rd\mu>0} at {s=0}, and, a fortiori, the pole of {\frac{1}{1-\lambda_s}} at {s=0} has order one.

Thus, the function (8) is analytic on {O_0-\{0\}}. Moreover, it also follows that the function (8) can be analytically extended to {O_0} if we show that

\displaystyle O_0\in s\mapsto \int\widehat{V}_s\cdot P_s(L_s\widehat{U}_{-s})

has a zero at {s=0}. This last fact is not hard to check: by definition, {P_0(L_0\widehat{U}_0) = c f_0} is a constant multiple of the function {f_0=d\mu/dLeb} and {\widehat{V}_0(x)=\int_0^{r(x)}V(x,a)da}, so that

\displaystyle \begin{array}{rcl} \int_{\Delta}\widehat{V}_0\cdot P_0(L_0\widehat{U}_{0})\, d Leb &=& c\int_{\Delta}\left(\int_0^{r(x)} V(x,a) \, da\right) f_0 \, d Leb \\ &=& c\int_{\Delta}\int_0^{r(x)} V(x,a) \, da\,d\mu := c\int_{\Delta_r} V \, d\mu_r \end{array}

and our assumption (2) was precisely that {\int V\,d\mu_r=0}.

This proves the lemma. \Box

Closing this post, let us reduce the step (c) to the following Dolgopyat-like estimate (compare with Proposition 7.7 in AGY paper):

Proposition 11 There exist {\sigma_0'>0}, {T_0\geq1}, {0<\beta<1} and {C>0} such that

\displaystyle \|L_s^ku\|_{L^2}\leq C\beta^k\|u\|_{1,t}

for all {s=\sigma+it} with {|\sigma|\leq\sigma_0'}, {|t|\geq T_0}, {k\in\mathbb{N}}. (Here, {\|u\|_{1,t}:=\|u\|_{C^0}+\frac{1}{|t|}\|Du\|_{C^0}} is the norm introduced in Remark 9.)

The proof of this proposition will occupy the next post of this series. For now, let us implement the step (c) of the Paley-Wiener strategy assuming Proposition 11.

We want to use the formula (4) to define a suitable analytic extension

\displaystyle \phi(s)=\sum\limits_{k=1}^{\infty}\int\widehat{V}_s\cdot L_s^k\widehat{U}_{-s}

of {\widehat{\rho}(s)} to a strip of the form {\{s=\sigma+it: |\sigma|<\sigma_0', |t|\geq T_0\}}.

By (6), Proposition 11, Corollary 7 and Lemma 8, we have

\displaystyle \begin{array}{rcl} |\phi(s)|&\leq& \sum\limits_{k=1}^{\infty}\|\widehat{V}_s\|_{L^2}\|L_s^{k-1}(L_s\widehat{U}_{-s})\|_{L^2} \\ &\leq& C\sum\limits_{k=1}^{\infty} \beta^{k-1}\|\widehat{V}_s\|_{L^2}\|L_s\widehat{U}_{-s}\|_{1,t} \\ &\leq& C\sum\limits_{k=1}^{\infty}\frac{\|V\|_{C^1}}{|t|}\frac{\|U\|_{C^1}}{|t|}\beta^{k-1} \\ &=& \frac{C}{(1-\beta)t^2}\|U\|_{C^1}\|V\|_{C^1} \end{array}

for all {s=\sigma+it} with {|\sigma|\leq\sigma_0'} and {|t|\geq T_0}.

This proves that {\phi(s)} is an analytic extension of {\widehat{\rho}(s)} to {\{s=\sigma+it: |\sigma|\leq\sigma_0', |t|\geq T_0\}} such that {|\phi(s)|\leq C/t^2}, which are exactly the properties required in the step (c).

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