FOR 2402/2: Rough Paths and Random Dynamical Systems (SP 04)

We investigate geometric properties of Weierstrass curves with one or two components, representing series based on trigonometric functions. They are Hölder continuous, and not controlled with respect to each other. They can be embedded into a smooth dynamical system, where their graph emerges as a pullback attractor. Each on-dimensional component of the curve may also be seen in the light of this dynamical system. It turns out that occupation measures and SBR measures on its stable manifold are dual to each other, via time reversal, A suitable version of approximate self similarity for deterministic functions yields approximate scaling properties for the measures., As a consequence, absolute continuity of the SBR measure is obtained, as well as the existence of a local time. The link between rough Weierstrass curves and smooth dynamical systems can be generalized considerably. Applications to regularization of singular ODE by rough (Weierstrass type) signals are on our agenda.