Rates for maps and flows in a deterministic multidimensional weak invariance principle

The work in this thesis concerns the branch of dynamics known as smooth ergodic theory. When a dynamical system and a probability measure are well-behaved, one can expect regular observables (typically Hölder continuous) to satisfy statistical properties that go beyond the classical ergodic theorem. Such results include the central limit theorem and its functional version, also called the weak invariance principle. The latter is analysed in the first (and main) part of this thesis, where we find rates of convergence to a Brownian motion in d-dimensions for discrete and continuous time systems. The proofs rely on a connection between dynamical system and martingale theory, via the martingale-coboundary decomposition introduced by Gordin [26]. The second part of the thesis presents results from two papers [42, 43] published by the author with Melbourne and Terhesiu, which discuss the decay of the transfer operator in continuous time. The chapter dedicated to [43] shows a restriction on the Banach spaces where such a transfer operator can have a spectral gap. The last chapter presents [42] and proves an exponential decay of the transfer operator in a strong norm for a class of nonuniformly expanding semiflows