The Non-multifractal Behavior for the Continuous Automorphism of the Torus (Cat) Map

Abstract

Multifractal analysis delves into the notion that different regions within a system can exhibit distinct scaling properties, challenging the traditional concept of uniform scaling. Unlike traditional fractals, which possess a single scaling exponent, multifractals capture the heterogeneity and self-similarity present in complex systems. This allows for a more nuanced understanding of the underlying dynamics, revealing hidden patterns and uncovering the intricate interplay between order and chaos.

A homogeneous set is referring to a set that is uniformly distributed. Which has no variations in density, structure, or other properties across its domain. The non-multifractal homogeneous set is a set that exhibits both non-multifractal properties and homogeneity. This means that the set displays a degree of self-similarity at a specific scale, while also having uniform distribution in its properties.

The paper considers the non-multifractal properties of a map of discrete-time dynamical systems from the 2-torus to itself and how uniformly distributed global stable and unstable manifolds can be.

This research employs the classical multifractal formalism to show the non-multifractal behavior of the Cat map, where the ergodicity of the map is explained. Then, theoretical multifractal functions are provided such as the cumulative curve length, the exponent , the fractal dimension , the dispersion , skew parameter , and the clustering coefficient . Lastly, the physical interpretations for the theoretical multifractal functions will be provided. Python data science is utilised to demonstrate the results.