In this thesis we consider discrete-time dynamical systems in the interval perturbed with bounded noise near a bifurcation of a minimal invariant set, its relation with stationary and quasi-stationary measures, and its asymptotic behaviour as the parameters tend to the bifurcation point.

First, we study existence and uniqueness of stationary measures on minimal invariant sets, and its approximation with the transition probabilities of the system. We derive analogous results for the unique stationary density, and conclude that the system exhibits exponential decay of (annhealed) correlations. We further study the shape of the stationary density near the boundary of its support, and relate it to the hyperbolicity of the the boundary point for the so-called extremal maps.

Secondly, we study a similar problem for quasi-stationary measures supported on a metastable set, when the parameters are close to a bifurcation point. We prove existence and uniqueness of quasi-stationary measures, and the exponential convergence of conditional transition probabilities of the system towards this measure. We study the asymptotic behaviour of the mean escape time from the metastable region as the parameter tends to the bifurcation point.

The results here presented relate the bifurcation theory of dynamical systems under bounded noisy perturbations, with their statistical features. To our knowledge these are among the first of their kind.