Cascade phenomenology

Lesson

Introduction
The matter of scale
Intermittency
Discrete cascade
Other type of cascade
- Introduction
- Auto-Similar spatial cascade
- Self-affine cascade
- Generalized scale invariance (GSI)
Exercises
To go further
Conclusion

Auto-Similar spatial cascade

The general principle of discrete cascade is the one of multiplicative cascade described in the previous section. The generated structure after some iteration are related to the initial structure by similarity of which the ratio of the generated structure on the initial one is .

These cascade are characterized by a perfect isotropic symmetry between the space dimension. The contraction-dilatation operator of the generator of these cascade, for the scale ration is given (2D) as :

$\begin{equation*} T_\lambda = \begin{pmatrix} \lambda^{-1} & 0 \\ 0 & \lambda^{-1} \end{pmatrix} \end{equation*}$

Which ca be expressed as :

$T_{\lambda} = \lambda^{-G}$

Where is the diagonal matrix expressed as :

$G = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

This definition allows us to observe the propriety of the scale invariance of the field. Indeed, the structure generator defined by the matrix being scale invariant, has an infinity of an invariant axis. We highlight the propriety of scale invariance is very import in the multifractal analysis developed in coming modules.

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