Entropy and decay of correlations for real analytic semi-flows
In the paper real analytic suspension semi-flows \(\Phi^t_{\tau}:X_{\tau} \rightarrow X_{\tau}\) over uniformly expanding real-analytic map of the interval are considered. It is shown that for any \(\Phi^t_{\tau}\)-invariant equilibrium measure \(\mu^{\tau}_g\) related to an analytic potential \(g\), there exists a Banach space \(\mathcal B_0 (g)\) of test functions such that for generic observables in \(\mathcal B_0 (g)\), the corresponding correlation functions \(Cor_g (F,G)(t)\) cannot decay faster than \(O(e^{-2h_g t})\), where \(h_g \) is the measure theoretic entropy of \(\mu^{\tau}_g\). This statement implies the existence of an essential spectrum for the Perron-Frobenius operator associated to the semi-flows, when acting on any reasonable Banach space.
scientific article; zbMATH DE number 5844004