Fractal Complex Dimensions and Cohomology of the Weierstrass Curve

International audience ; In this survey article, we present the authors' main results concerning the Complex Dimensions of the Weierstrass Curve, along with their links with the associated fractal cohomology, as developed in our previous papers [DL22a], [DL22b]. Our results shed new light on the theory and the interpretation of Complex Fractal Dimensions, insofar as we envision the fractal Complex Dimensions as dynamical quantities, which evolve with the scales. Accordingly, we define the Complex Dimensions of the Weierstrass Curve as the set of the Complex Dimensions of the sequence of Weierstrass Relative Fractal Drums which converge to the Curve. By means of fractal tube formulae, we then obtain the associated Weierstrass fractal zeta functions, whose poles yield the set of Complex Dimensions. In particular, we show that the Complex Dimensions (apart from 0 and −2) are periodically distributed along countably many vertical lines, with the same oscillatory period. As expected, the Minkowski (or box-counting) dimension is the Complex Dimension with maximal real part, and zero imaginary part. We then show how those Dimensions are connected to the cohomological properties of the Curve: the cohomological groups related to the Curve are obtained, by induction, as sums indexed by the cohomological Complex Dimensions. We determine explicitly both the infinite sequence of prefractal cohomology spaces and the corresponding inductive limit, the fractal (or total) cohomology space of the Weierstrass Curve. In particular, we show that the elements of these cohomology spaces-viewed as suitable continuous functions on the Curve-admit a fractal power series expansion taken over the cohomological Complex Dimensions, that are akin to Taylor-like expansions.