Toward a deeper understanding of a basic cascade

The study of turbulence and chaos has led to crucial new concepts. In particular, B. Mandelbrot introduced and promoted (multi-)fractals. More recently, cascades have gained momentum. Not least due to technical difficulties, continuous stochastic models, such as the classic p model, have been preferred over discrete cascades. It is the aim of this contribution to introduce original concepts that shed new light on a variant of the latter paradigmatic setting and allow key features to be derived in a rather elementary fashion. To this end, we introduce and study a discrete version of the p model that is based on a new kind of sampling, named "power sampling." Technical machinery can be kept simple; therefore, proofs are straightforward and formulas explicit. It is hoped that the proposed line of investigation may enhance understanding and simplify received multi-fractal analysis.