Is there always a conservation law behind the emergence of fractal and multifractal?

Abstract

One of the most basic ingredients of fractal or multifractal is its scale-invariance or self-similar property albeit they appear seemingly disordered or apparently bewildering. In this article, we give several examples of deterministic, random and stochastic fractals as well as multifractals to show that there is always at least one conservation law behind all these systems. On the other hand, it is well-known and it has been shown here too that fractals and multifractals are self-similar which is also some form of symmetry that sends the object to itself. This is reminiscent to Noether’s first theorem that states that for every continuous symmetry of an action, there exists a conserved quantity. Finding the connection between conserved quantity in fractal and multifractal with scale-invariance of self-similarity can actually be coined as an equivalent counterpart of the Noether’s first theorem.