The pressure and higher correlations for an Anosov diffeomorphism (Q2730713)

Let \((X,\varphi)\) be an Anosov diffeomorphism with Gibbs measure \(\mu=\mu_f\). In this paper, the authors extend Ruelle's results related to the exponential decay of correlation and the relation between correlation and the pressure of the dynamical system. NEWLINENEWLINENEWLINEDefine the higher covariance of functions \(f_0,\ldots,f_k\) by NEWLINE\[NEWLINE\text{Cov}(f_0,\ldots,f_k) =\left.{\partial^{k+1}\over\partial\theta_0\cdots\partial\theta_k} \log\left(\int\exp\left(\sum_{i=0}^k\theta_if_i\right) d\mu\right) \right|_{\theta_0=\cdots=\theta_k=0},NEWLINE\]NEWLINE then the higher correlation function is expressed by NEWLINE\[NEWLINE\rho_{f_0,\ldots,f_k}(n_1,\ldots,n_k) =\text{Cov}(f_0,f_1\circ\varphi^{n_1},\ldots,f_k\circ\varphi^{n_k}).NEWLINE\]NEWLINE The first theorem is: There exist constants \(C>0\) and \(\lambda\in(0,1)\) so that NEWLINE\[NEWLINE\rho_{f_0,\ldots,f_k}(n_1,\ldots,n_k)\leq C\lambda^{|n_1|+\cdots+|n_k|}.NEWLINE\]NEWLINE Let us denote by \(p(f)\) the pressure of the dynamical system, and define NEWLINE\[NEWLINE\delta_f^{k+1}(f_0,\ldots,f_k)= \left.{\partial^{k+1}\over\partial t_0\cdots\partial t_k} p(f+t_0f_0+\cdots+t_kf_k)\right|_{t_0=\cdots=t_k=0}.NEWLINE\]NEWLINE Then the second theorem is: NEWLINE\[NEWLINE\delta_f^{k+1}(f_0,\ldots,f_k)=\sum_{n_1,\ldots,n_k\in Z} \rho_{f_0,\ldots,f_k}(n_1,\ldots,n_k).NEWLINE\]NEWLINE For \(I=\{i_1,\ldots,i_s\}\subset\{0,\ldots,k\}\), define NEWLINE\[NEWLINEf_I=f_{i_1}\cdots f_{i_s},\qquad \mathbf f_I=(f_{i_1},\ldots, f_{i_s}).NEWLINE\]NEWLINE Then for a partition \(P=\{I_1,\ldots,I_l\}\) of \(\{0,\ldots,k\}\), define NEWLINE\[NEWLINE\langle \mathbf f\rangle_P=\langle \mathbf f_{I_1}\rangle\cdots\langle \mathbf f_{I_l}\rangle,NEWLINE\]NEWLINE where \(\langle f\rangle=\int f d\mu\). Now consider \(\langle \mathbf f\rangle_P\) as formal symbol, and introduce \(\text{Cov}^0(\mathbf f_I)\), which is similar to \(\text{Cov}(f)\). Then \(\text{Cov}(\mathbf f_P)\) can be expressed by a linear combination of \(\text{Cov}^0(\mathbf f_I)\). NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINENow express the dynamical system by symbolic dynamics \((\Sigma_A,\sigma)\), and a one-sided sequence space \(\Sigma_A^+\). On these symbolic dynamics, one considers as usual Banach spaces \({\mathcal F}_\theta\) and \({\mathcal F}_\theta^+\) \((0<\theta<1)\), and the Ruelle-Perron-Frobenius operator on \({\mathcal F}_\theta^+\). Then using the spectral theory of this operator, there exist constants \(C>0\) and \(0<\lambda<1\) such that NEWLINE\[NEWLINE|\text{Cov}^0(f_{i_1}^{n_{i_1}}.\ldots,f_{i_s}^{n_{i_s}})|\leq \lambda^{n_{i_s}-n_{i_1}}||f_{i_1}||_\theta\cdots||f_{i_s}||_\theta.NEWLINE\]NEWLINE From this inequality, one can prove Theorem 1. Theorem 2 can be proved by induction.