Operator renewal theory for continuous time dynamical systems with finite and infinite measure

Abstract

We develop operator renewal theory for flows and apply this to obtain results on mixing and rates of mixing for a large class of finite and infinite measure semiflows. Examples of systems covered by our results include suspensions over parabolic rational maps of the complex plane, and nonuniformly expanding semiflows with indifferent periodic orbits. In the finite measure case, the emphasis is on obtaining sharp rates of decorrelations, extending results of Gouëzel and Sarig from the discrete time setting to continuous time. In the infinite measure case, the primary question is to prove results on mixing itself, extending our results in the discrete time setting. In some cases, we obtain also higher order asymptotics and rates of mixing.

Wiener lemma

This appendix contains material about a version of the Wiener lemma that is required in Sect. 13. We have chosen the notation here to conform with standard conventions in Fourier analysis. (In the application of this material, the roles of $f:{\mathbb R}\rightarrow {\mathbb C}$ and its Fourier transform $\hat{f}$ is reversed, with b and t playing the role of x and $\xi$ respectively.)

Let ${\mathbb A}$ be the Banach algebra of $2\pi$-periodic continuous functions $f:{\mathbb R}\rightarrow {\mathbb C}$ such that their Fourier coefficients $\hat{f}_n$ are absolutely summable, with norm $\Vert f\Vert _{{\mathbb A}}=\sum _{n\in {\mathbb Z}}|\hat{f}_n|$. Similarly, let $\mathcal {R}$ be the Banach algebra of continuous functions $f:{\mathbb R}\rightarrow {\mathbb C}$ such that their Fourier transform $\hat{f}:{\mathbb R}\rightarrow {\mathbb C}$ lies in $L^1({\mathbb R})$, with norm $\Vert f\Vert _{\mathcal {R}}=\int _{-\infty }^\infty |\hat{f}(\xi )|\,d\xi$.

Given $\beta >1$, we define the Banach algebra ${\mathbb A}_\beta =\{f\in {\mathbb A}:\sup _{n\in {\mathbb Z}}|n|^\beta |\hat{f}_n|<\infty \}$ with norm $\Vert f\Vert _{{\mathbb A}_\beta }=\sum _{n\in {\mathbb Z}}|\hat{f}_n|+\sup _{n\in {\mathbb Z}}|n|^\beta |\hat{f}_n|$. Similarly, we define the Banach algebra $\mathcal {R}_\beta =\{f\in \mathcal {R}:\sup _{\xi \in {\mathbb R}}|\xi |^\beta |\hat{f}(\xi )|<\infty \}$ with norm $\Vert f\Vert _{\mathcal {R}_\beta }=\int _{-\infty }^\infty |\hat{f}(\xi )|\,d\xi +\sup _{\xi \in {\mathbb R}}|\xi |^\beta |\hat{f}(\xi )|$.

The following Wiener lemmas are standard.

Lemma 16.1

Let $\beta >1$ and let $f,f_1\in \mathcal {A}_\beta$. Suppose that f is bounded away from zero on the support of $f_1$.

Then there exists $g\in \mathcal {A}_\beta$ such that $f_1=fg$.

Lemma 16.2

Let $\beta >1$ and let $f,f_1\in \mathcal {R}_\beta$. Suppose $f_1$ is compactly supported and that f is bounded away from zero on the support of $f_1$.

Then there exists $g\in \mathcal {R}_\beta$ such that $f_1=fg$.

A statement and proof of Lemma 16.1 can be found in [13, Theorem 1.2.12]. In this paper, we require Lemma 16.2, but we could not find it stated in the literature. Hence we provide here a proof of Lemma 16.2, using a standard argument to reduce to Lemma 16.1.

Lemma 16.3

Let $\epsilon >0$. Suppose that $f:{\mathbb R}\rightarrow {\mathbb C}$ is a continuous function with ${\text {supp}}f\subset [-\pi +\epsilon ,\pi -\epsilon ]$. Let $h:{\mathbb R}\rightarrow {\mathbb C}$ denote the $2\pi$-periodic continuous function such that $h|_{[-\pi ,\pi ]}=f|_{[-\pi ,\pi ]}$. Then $f\in \mathcal {R}_\beta$ if and only if $h\in {\mathbb A}_\beta$.

Proof

(cf. [21, Theorem 6.2, Ch. VIII, p. 242]) Fix a $C^\infty$ function $\psi :{\mathbb R}\rightarrow {\mathbb R}$ supported in $[-\pi +\epsilon /2,\pi -\epsilon /2]$ and such that $\psi \equiv 1$ on $[-\pi +\epsilon ,\pi -\epsilon ]$. For $\alpha \in [-1,1]$ let $\psi _\alpha (x)=e^{i\alpha x}\psi (x)$. Then there is a constant $K_0>0$ such that

$$\begin{aligned} |(\widehat{\psi }_\alpha )_n|\le K_0n^{-\beta },\quad \text {for all}\quad \alpha \in [-1,1], n\in {\mathbb Z}. \end{aligned}$$

In particular, $\psi _\alpha \in {\mathbb A}_\beta$ for all $\alpha$ and $\sup _{|\alpha |\le 1}\Vert \psi _\alpha \Vert _{{\mathbb A}_\beta }<\infty$.

Define $h_\alpha (x)=e^{i\alpha x}h(x)$. If $h\in {\mathbb A}_\beta$, then $h_\alpha =h\psi _\alpha \in {\mathbb A}_\beta$ and there is a constant $K>0$ such that $\Vert h_\alpha \Vert _{{\mathbb A}_\beta }\le K\Vert h\Vert _{{\mathbb A}_\beta }$ for all $\alpha \in [-1,1]$.

Now,

$$\begin{aligned} (\widehat{h_\alpha })_n=\frac{1}{2\pi }\int _{-\pi }^\pi e^{i\alpha x}h(x)e^{-inx}\,dx= \frac{1}{2\pi }\int _{-\infty }^\infty f(x)e^{-i(n-\alpha ) x}\,dx= \frac{1}{2\pi }\hat{f}(n-\alpha ). \end{aligned}$$

Hence $\int _{n-1}^n|\hat{f}(\xi )|\,d\xi =\int _0^1|\hat{f}(n-\alpha )|\,d\alpha =2\pi \int _0^1|(\widehat{h_\alpha })_n|\,d\alpha$. It follows that

$$\begin{aligned} \Vert f\Vert _{\mathcal {R}}=2\pi \sum _{n=-\infty }^\infty \int _0^1 |(\widehat{h_\alpha })_n|\,d\alpha =2\pi \int _0^1\Vert h_\alpha \Vert _{{\mathbb A}}\,d\alpha \le 2\pi K\Vert h\Vert _{{\mathbb A}_\beta }. \end{aligned}$$

(16.1)

Next, we observe that any $\xi \in {\mathbb R}$ can be expressed as $\xi =(n-\alpha ){\text {sgn}}\xi$ where $n\ge 1$, $\alpha \in [0,1]$. Hence

$$\begin{aligned} \nonumber&\sup _{\xi \in {\mathbb R}}|\xi |^\beta |\hat{f}(\xi )|\nonumber \\&\quad =\sup _{n\ge 1,\,\alpha \in [0,1]} (n-\alpha )^\beta |\hat{f}((n-\alpha ){\text {sgn}}\xi )| \le \sup _{n\ge 1,\,\alpha \in [0,1]} n^\beta |\hat{f}((n-\alpha ){\text {sgn}}\xi )| \nonumber \\&\quad \le \sup _{n\in {\mathbb Z},\,\alpha \in [-1,1]} |n|^\beta |\hat{f}(n-\alpha )| = 2\pi \sup _{n\in {\mathbb Z},\,\alpha \in [-1,1]} |n|^\beta |(\widehat{h_\alpha })_n| \nonumber \\&\quad \le 2\pi \sup _{\alpha \in [-1,1]} \Vert h_\alpha \Vert _{A_\beta } \le 2\pi K\Vert h\Vert _{A_\beta }. \end{aligned}$$

(16.2)

Combining (16.1) and (16.2), we obtain that $\Vert f\Vert _{\mathcal {R}_\beta }\le 4\pi K\Vert h\Vert _{A_\beta }$. Hence we have shown that $h\in {\mathbb A}_\beta$ implies that $f\in \mathcal {R}_\beta$.

Conversely, suppose $f\in \mathcal {R}_\beta$. Then $\sum _{n\in {\mathbb Z}}\int _0^1|\hat{f}(n-\alpha )|\,d\alpha =\int _{-\infty }^\infty |\hat{f}(\xi )|\,d\xi <\infty$ and it follows from Fubini that $\sum _{n\in {\mathbb Z}}|\hat{f}(n-\alpha )|<\infty$ for almost every $\alpha$. Fix such an $\alpha$. Then $\sum _{n\in {\mathbb Z}}|(\widehat{h_\alpha })_n|=(1/2\pi ) \sum _{n\in {\mathbb Z}}|\hat{f}(n-\alpha )|<\infty$ so that $h_\alpha \in {\mathbb A}$. Hence $h=(h_\alpha )_{-\alpha }\in {\mathbb A}$. Moreover,

$$\begin{aligned} \sup _{n\in {\mathbb Z}}|n|^\beta |\hat{h}_n| =(1/2\pi ) \sup _{n\in {\mathbb Z}}|n|^\beta |\hat{f}(n)| \le (1/2\pi ) \sup _{\xi \in {\mathbb R}}|\xi |^\beta |\hat{f}(\xi )|<\infty , \end{aligned}$$

so that $h\in {\mathbb A}_\beta$. $\square$

Proof of Lemma 16.2

(cf. [21, Lemma 6.3, Ch. VIII, p. 242]) We make the standard abuse of notation that functions on ${\mathbb R}$ supported on a closed subset of $(-\pi ,\pi )$ can be identified with $2\pi$-periodic functions on ${\mathbb R}$. In particular, the conclusion of Lemma 16.3 becomes $f\in \mathcal {R}_\beta$ if and only if $f\in {\mathbb A}_\beta$.

Without loss, we can suppose that ${\text {supp}}f_1\subset [-2,2]$. By Lemma 16.3, $f_1\in {\mathbb A}_\beta$.

Choose a $C^\infty$ function $\chi :{\mathbb R}\rightarrow {\mathbb R}$ such that ${\text {supp}}\chi \subset [-3,3]$ and $\chi \equiv 1$ on $[-2,2]$. Then $\chi \in \mathcal {A}_\beta$ and $\chi \in \mathcal {R}_\beta$. In particular $\chi f\in \mathcal {R}_\beta$, and by Lemma 16.3 $\chi f\in {\mathbb A}_\beta$.

Moreover $\chi f=f$ on ${\text {supp}}f_1$ and hence is bounded away from zero on ${\text {supp}}f_1$. By Lemma 16.1, there exists $g_0\in {\mathbb A}_\beta$ such that $f_1=g_0(\chi f)=(g_0\chi )f$.

Since $g_0,\chi \in {\mathbb A}_\beta$, we deduce that $g=g_0\chi \in {\mathbb A}_\beta$. By Lemma 16.3, $g\in \mathcal {R}_\beta$. Hence $f_1=gf$ with $g\in \mathcal {R}_\beta$ as required.