Multifractals and Turbulence

Part 1 Historical background and scientific context: from speculative turbulance to phenomenological cascades models; scientific crisis; symmetries and complexity; scale symmetry; fields and geometry. Part 2 Hydrodynamic turbulence: equations of evolution; basic physics of three-dimensional turbulence; symmetries and conservation laws; statistical descriptors and singular behaviours; the very special case of 2-D turbulence; some mathematical exact results and conjectures; unpredictability and the caricature of deterministic chaos; renormalization techniques. Part 3 Intermittency: failures of homogeneous turbulence; phenomenological cascades; discrete multiplicative processes; dual representations - singularities/statistical moments; continuous multiplicative processes. Part 4 Multifractal processes: general considerations; classification of the multifractal singularities and processes; bare and dressed properties; universality; miscellaneous properties; fluxdynamics and thermodynamics formalisms; renormalization revisited; data analysis methods; numerical simulations. Part 5 Generalized scale invariance: scaling anisotropy in space and in space-time; global scale transformations; local scale transformation; stochastic scale transformation; data analysis methods; numerical simulations.