Cutting-Decimation Renormalization for diffusive and vibrational dynamics on fractals

TL;DR

This paper investigates how non exactly decimable fractals exhibit a phenomenon called dynamical dimension splitting, where diffusive and vibrational dynamics require separate parameters, due to their invariance under a complex cutting-decimation transform.

Contribution

It provides a detailed analysis of the mathematical properties of the cutting-decimation transform and explains the origin of dynamical dimension splitting in non exactly decimable fractals.

Findings

Dynamical dimension splitting arises from the cutting-decimation transform.

Exact decimability leads to dynamical dimension degeneration.

The study clarifies the role of invariance under complex transforms in fractal dynamics.

Abstract

Recently, we pointed out that on a class on non exactly decimable fractals two different parameters are required to describe diffusive and vibrational dynamics. This phenomenon we call dynamical dimension splitting is related to the lack of exact decimation invariance for these structures, which turn out to be invariant under a more complex cutting-decimation transform. In this paper we study in details the dynamical dimension splitting on these fractals analyzing the mathematical properties of the cutting-decimation transform. Our results clarify how the splitting arises from the cutting transform and show that the dynamical dimension degeneration is a very peculiar consequence of exact decimability.