When a dynamical system admitting a natural (SRB) measure is perturbed, it is natural to ask how the SRB measure responds to the perturbation. In the tamest cases, this response is linear, and the derivative of the SRB measure with respect to the parameter can be expressed as a sum of decorrelations (involving the derivative of the system with respect to the parameter). In more subtle situations - for example, systems with bifurcations, or observables with singularities - the SRB measure may be a Hölder function of the parameter. This talk will present a panorama of results about linear and fractional response.[-]

The SRB measure of Sinai billiard maps and flows has been studied for decades, but other equilibrium states have been investigated only recently. Assuming finite horizon, the measure of maximal entropy (MME) of the (discontinuous) map has been constructed and shown to be unique and Bernoulli (joint work with Demers, 2020), under a mild condition (believed to be generic) on the topological entropy. Demers and Korepanov have recently shown that this MME mixes at least polynomially (for H¨older observables). In spite of the continuity of the billiard flow, the mere existence of the MME for the flow has been a challenging problem. I will explain how we obtain existence, uniqueness and Bernoullicity of the MME of the Sinai billiard flow, assuming finite horizon and a mild condition (also believed to be generic), by bootstrapping on very recent work of J´erˆome Carrand about a family of equilibrium states for the billiard map. We use transfer operators acting on anisotropic Banach spaces. (Joint work with J´erˆome Carrand and Mark Demers).[-]