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Subexponential decay of correlations. (English) Zbl 1042.37005
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 $$\operatorname{Hom}({\mathcal L},{\mathcal L})$$ for every $$| z| <1$$. Assume the
Renewal equation: for every $$| z| <1$$, $$T(z)=(I-R(z))^{-1}$$ where $$R(z)=\sum_{n\geq1}z^nR_n$$, $$R_n\in\operatorname{Hom}({\mathcal L},{\mathcal L})$$ 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 $$E_n\in\operatorname{Hom}({\mathcal L},{\mathcal L})$$ 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 Hölder 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 Hölder 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.

##### MSC:
 37A30 Ergodic theorems, spectral theory, Markov operators 28D05 Measure-preserving transformations 47A35 Ergodic theory of linear operators 37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
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