Dynamical zeta functions for Anosov flows via microlocal analysis (Q2811187)

In this article, the meromorphic continuation of the Ruelle zeta function for smooth Anosov flows on compact manifolds is proven by means of microlocal analysis.NEWLINENEWLINELet \(X\) be a compact manifold and \(\varphi_t:X\to X\) a \(C^\infty\) Anosov flow with orientable stable and unstable bundles. Let \(\{\gamma^\#\}\) denote the set of primitive periodic orbits of the flow and \(T_\gamma^\#\) their periods. Then the Ruelle zeta function is defined for Im\((\lambda)\gg1\) as NEWLINE\[NEWLINE \zeta_R(\lambda)=\prod_{\gamma^\#}(1-e^{i\lambda T_\gamma^\#}). NEWLINE\]NEWLINE The main result of this article shows that \(\zeta_R(\lambda)\) has a meromorphic continuation to \(\mathbb C\). This result has been obtained previously in [\textit{P. Giulietti} et al., Ann. Math. (2) 178, No. 2, 687--773 (2013; Zbl 1418.37042)] with a different spectral theoretical approach.NEWLINENEWLINEIn the present article the outline of the proof is the following: {\parindent=0.7cm \begin{itemize}\item[--] First the authors study the differential operator \(\mathbf{P}=\frac{1}{i}\mathcal L_V\) where \(V\) is the vector field of the Anosov flow and \(\mathcal L_V\) the corresponding Lie derivative on differential forms. They prove (Section 3.2) that the resolvent \((\mathbf{P} - \lambda)^{-1}\) has a meromorphic extension to any half plane \(\{\text{Im}(\lambda)>-C_0\}\) as an operator acting on certain anisotropic Sobolev spaces. The definition of these anisotropic Sobolev spaces is very similar to the construction in [\textit{F. Faure} and \textit{J. Sjöstrand}, Commun. Math. Phys. 308, No. 2, 325--364 (2011; Zbl 1260.37016)], however the techniques for proving the meromorphic continuation differ. The present article uses a semiclassical propagation of singularities and radial sink and source estimates. These techniques allow to give a precise description of the wavefront set of the continued resolvent operator. This is a new and interesting result (Proposition 3.3). \item[--] The wavefront set of \((\mathbf{P} - \lambda)^{-1}\) allows one to take the flat trace of this resolvent acting on \(k\)-forms that are anihilated by \(V\). Via these flat traces the authors introduce a spectral zeta function \(\zeta_k(\lambda)\). By the meromorphic continuation of the resolvent all these zeta functions are analytic functions and using Guillemin's trace formula, they can be expressed as sums over the periodic orbits. \item[--] Finally, via a trace identity, \(\zeta_R(\lambda)\) can be expressed as an alternating product over \(\zeta_k(\lambda)\) which establishes the meromorphic continuation of the Ruelle zeta function. NEWLINENEWLINE\end{itemize}}