an:01965454
Zbl 1042.37005
Sarig, Omri
Subexponential decay of correlations
EN
Invent. Math. 150, No. 3, 629-653 (2002).
00089506
2002
j
37A30 28D05 47A35 37C30
transfer operator; renewal equation; measure-preserving irreducible Markov dynamical system
Let \((X,{\mathcal B},m,T)\) be a dynamical system, and denote by \(\widehat T\) the transfer operator (Perron-Frobenius operator) associated with the dynamical system. The first assertion is the following renewal equation.
Proposition 1. Let the dyamical system be conservative and nonsingular. Assume that \(A\in{\mathcal B}\) has finite positive measure. Let \(T_nf=1_A\widehat T^n(f1_A)\) and \(R_nf=1_A\widehat T^n(f1_{[\phi_A=n]})\), where \(\phi_A(x)\) is the first return time to \(A\) for \(x\in A\). Then for any \(| z| <1\)
\[
T(z)=(I-R(z))^{-1},\quad R(z)=\sum_{n=1}^\infty a^nR_n,\quad T(z) =\sum_{n=0}^\infty z^nT_n.
\]
The main result of this article is:
Theorem 1. Let \(T_n\) be bounded linear operators on a Banach space \(\mathcal L\) such that \(T(z)=\sum_{n=0}^\infty z^nT_n\) converges in \(\Hom({\mathcal L},{\mathcal L})\) for every \(| z| <1\). Assume the
Renewal equation: for every \(| z| <1\), \(T(z)=(I-R(z))^{-1}\) where \(R(z)=\sum_{n\geq1}z^nR_n\), \(R_n\in\Hom({\mathcal L},{\mathcal L})\) and \(\sum\| R_n\| <\infty\).
Spectral gap: the spectum of \(R(1)\) consists of an isolated simple eigenvalue at \(1\) and a compact subset in \(| z| <1\).
Aperiodicity: the spectral radius of \(R(z)\) is strictly less than one for all \(| z| \leq 1\) and \(z\neq1\).
Let \(P\) be the eigenprojection of \(R(1)\) at \(1\).
If \(\sum_{k>n}\| R_k\| =O(1/n^\beta)\) for some \(\beta>2\) and \(PR'(1)\neq0\), then for all \(n\)
\[
T_n={1\over\mu}P+{1\over\mu^2}\sum_{k=n+1}^\infty P_k+E_n
\]
where \(\mu\) is given by \(PR'(1)P=\mu P\), \(P_n=\sum_{l>n}PR_lP\), and \(E_n\in\Hom({\mathcal L},{\mathcal L})\) satisfy \(\| E_n\| =O(1/n^{[\beta]})\).
Theorem 2 gives lower bounds for the correlation functions \(s\):
For a Markov partition \(\alpha\), let \([a_,\ldots,a_{n-1}]\) be the cylinder generated by \(a_1,\ldots,a_{n-1}\in\alpha\). Let
\[
v_n(\phi)=\sup\{| \phi(x)-\phi(y)| : x,y\in[a_0,\ldots,a_{n-1}],a_i\in\alpha\}.
\]
The dynamical system is called \((T,\alpha)\)-summable if \(\sum_{n\geq2}v_n(\phi)<\infty\), and \((T,\alpha)\)-locally H??lder continuous if there exists \(A>0\), \(\theta\in(0,1)\) such that \(v_n(\phi)<A\theta^n\) for all \(n\). Let \(T_a\) be a induced transformation on \(a\in\alpha\), and the Markov partition induced on \(a\) is denoted by \(\alpha_a\). Moreover
\[
\begin{aligned} g_m&={dm\over dm\circ T},\\ s(x,y)&=\sup\{n\geq0: x,y\in[b_0,\ldots,b_{n-1}],b_i\in\alpha_a\},\\ D_af&=\sup| f(x)-f(y)| /\theta^{s(x,y)}. \end{aligned}
\]
Define
\[
\| f\| _{\mathcal L}=\| f\| _\infty+D_af.
\]
Then Theorem 2. For a measure-preserving, irreducible Markov dynamical system, assume that \(\log g_{m_a}\) has \((T_a,\alpha_a)\)-locally H??lder continuous version for some \(a\in\alpha\). If g.c.d. \(\{\phi_a(x)-\phi-a(y): x,y\in\cup\alpha_a\}=1\) and \(m[\phi_a>n]=O(1/n^\beta)\) where \(\beta>2\), then there exists \(\theta\in(0,1)\) \(C>0\) such that for any \(f,g\) integrable supported inside \([a]\),
\[
\left| \text{Cor}(f,g\circ T^n)- \left(\sum_{k=n+1}^\infty m[\phi_a>k]\right) \int f\int g\right| \leq Cn^{-[\beta]}\| g\| _\infty\| f\| _{\mathcal L}.
\]
Two examples are given and correlations of them are calculated. One is an extension of the Mannville-Pomeau map, and the other are LS Young towers.
Makoto Mori (Tokyo)