On scale invariance, multifractal formalism, and their applications

Scale invariance represents a relatively new way to analyse and model real world data. It finds numerous applications in many diverse fields, including the study of natural phenomena in physics, biology, but also man-made applications such as signal and image processing. Data presenting scale invariance do not have a well defined time scale playing a dominant role. Instead, their dynamics can be understood from the relationship existing across a whole range of scales. Multifractal theory is particularly useful for analysing this kind of data. In this talk I will present some basic notions of fractal theory, and then discuss the multifractal formalism.
I will illustrate the theory with examples taken from the literature and my research, including multiplicative cascades on trees and random processes in multifractal time.