Lin\'earisation d'une it\'eration born\'ee dans Rd par des fonctions de Weierstrass

TL;DR

This paper demonstrates that semi-invariant curves of a bounded polynomial diffeomorphism iteration in Rd asymptotically resemble Weierstrass functions, providing a fractal and self-similar framework applicable to differential calculus.

Contribution

It introduces a novel connection between bounded polynomial diffeomorphism iterations and Weierstrass functions, justifying self-similarity and fractal analysis in this context.

Findings

Semi-invariant curves tend to Weierstrass parametrized curves.

Fractal dimension and self-similarity are justified for these curves.

Results are applied to partial differential calculus.

Abstract

In this paper, we study an iteration in defined by a diffeomorphism polynomial bounded. Semi invariant curves tend to curves with parametric Weierstrass-Mandelbrot's functions. So, self-similarity and fractal dimension are justified. We apply these results to partial differential calculus. On \'etudie une it\'eration de Rd dans Rd d\'efinie par un diff\'eomorphisme polynomial born\'e. On montre que les courbes semi invariantes tendent asymptotiquement vers des courbes param\'etr\'ees par des fonctions de Weierstrass. Cela justifie les calculs d'\'echelle d'autosimilarit\'e et de dimension fractale comme le pratiquent des praticiens \`a partir d'intuitions pertinentes sur des it\'erations chaotiques. On applique ces r\'esultats au calcul diff\'erentiel.