Resolvent of vector fields and Lefschetz numbers

Abstract

Dynamical series such as the Ruelle zeta function have become a staple in the study of hyperbolic flows. They are usually analyzed by relating them to the resolvent of the vector field. In this paper we give the general form of such relations, which involves the intersection of the kernel of said resolvent with integration currents. Our formula is actually valid for any smooth flow, not necessarily hyperbolic. As an application, we introduce certain dynamical series that had not appeared before. Finally, we compute their value at zero, and their relation with topological invariants.

References in corpus (7)

• Upper bound on the density of Ruelle resonances for Anosov flows
• Ruelle zeta function at zero for surfaces
• Control of eigenfunctions on surfaces of variable curvature
• Fried conjecture in small dimensions
• Dynamical zeta functions for Axiom A flows
• Analytic torsion, dynamical zeta function, and the Fried conjecture for admissible twists
• Poincaré series for surfaces with boundary