Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems

Abstract

AbstractIn this article we prove meromorphic continuation of weighted zeta functions $$Z_f$$ Z f in the framework of open hyperbolic systems by using the meromorphically continued restricted resolvent of Dyatlov and Guillarmou (Ann Henri Poincaré 17(11):3089–3146, 2016). We obtain a residue formula proving equality between residues of $$Z_f$$ Z f and invariant Ruelle distributions. We combine this equality with results of Guillarmou et al. (Ann Henri Lebesgue 4:81–119, 2021) in order to relate the residues to Patterson–Sullivan distributions. Finally we provide proof-of-principle results concerning the numerical calculation of invariant Ruelle distributions for 3-disc scattering systems.