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We establish a new relation between classical observables for scattering andbound orbits of a massive probe particle in a Kerr background. We find an exactrepresentation of the Hamilton-Jacobi action in terms of the conserved chargeswhich admits an analytic continuation, both for the radial and polarcontribution, for a general class of geodesics beyond the equatorial case.Remarkably, this allows to extend the boundary to bound dictionary and itprovides an efficient method to compute the deflection angles and the timedelay for scattering orbits, as well as frequency ratios for bound orbits, inthe probe limit but at all orders in the perturbative expansion.

Boundary to bound dictionary for generic Kerr orbits