  We consider the general response theory proposed by Ruelle for describing the
impact of small perturbations to the non-equilibrium steady states resulting
from Axiom A dynamical systems. We show that the causality of the response
functions allows for writing a set of Kramers-Kronig relations for the
corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only
a special class of observable susceptibilities obey Kramers-Kronig relations.
Specific results are provided for arbitrary order harmonic response, which
allows for a very comprehensive Kramers-Kronig analysis and the establishment
of sum rules connecting the asymptotic behavior of the susceptibility to the
short-time response of the system. These results generalize previous findings
on optical Hamiltonian systems and simple mechanical models, and shed light on
the general impact of considering the principle of causality for testing
self-consistency: the described dispersion relations constitute unavoidable
benchmarks for any experimental and model generated dataset. In order to
connect the response theory for equilibrium and non equilibrium systems, we
rewrite the classical results by Kubo so that response functions formally
identical to those proposed by Ruelle, apart from the measure involved in the
phase space integration, are obtained. We briefly discuss how these results,
taking into account the chaotic hypothesis, might be relevant for climate
research. In particular, whereas the fluctuation-dissipation theorem does not
work for non-equilibrium systems, because of the non-equivalence between
internal and external fluctuations, Kramers-Kronig relations might be more
robust tools for the definition of a self-consistent theory of climate change.
