Stripping singularities accelerates cycle expansions

Abstract The convergence of cycle expansions in non-hyperbolic systems has long been an outstanding challenge, primarily due to the presence of singularities in natural measures and the attributed weak shadowing properties of unstable periodic orbits. Here, we introduce a novel layered approximation scheme that effectively decomposes the natural measure into distinct singular and hyperbolic layers. For the singular components, we employ analytical approximations, whereas for the hyperbolic components, we utilize cycle expansions. A key innovation of our approach is the incorporation of dynamical information beyond periodic orbits, which is essential for achieving a more accurate approximation of the natural measure and for precisely calculating observable averages. By integrating additional dynamical details, the new scheme more effectively captures the subtleties of the system’s behavior that are not fully represented by a limited set of periodic orbits alone. This invariance allows us to refine the approximation of the natural measure in both the singular and the hyperbolic layers, thereby yielding a more reliable estimation of observable averages.