We investigate geometric properties of Weierstrass curves with two components, representing series based on trigonometric functions. They are seen to be $\frac12$-Hölder continuous, and are not (para-)controlled with respect to each other in the sense of the recently established Fourier analytic approach of rough path analysis. Their graph is represented as an attractor of a smooth random dynamical system. Our argument that its graph has Hausdorff dimension 2 is in the spirit of Ledrappier-Young’s approach of the Hausdorff dimension of attractors. This is joint work with G. dos Reis (U Edinburgh) and O. Pamen (U Liverpool and AIMS Ghana).