# Abstract Similarity, Fractals and Chaos

## TL;DR

The paper introduces a new concept of abstract similarity to establish chaos in fractal spaces, demonstrating its application to classic fractals and symbolic string spaces through mathematical proofs and numerical simulations.

## Contribution

It presents a novel mathematical framework of abstract similarity to analyze chaos in fractals, extending to multi-dimensional cases and providing new insights into fractal dynamics.

## Key findings

- Fractals like Sierpinski, Koch, and Cantor satisfy the abstract similarity definition.
- Chaos types such as Poincare, Li-Yorke, and Devaney are confirmed in fractal dynamics.
- Numerical simulations support the theoretical results.

## Abstract

To prove presence of chaos for fractals, a new mathematical concept of abstract similarity is introduced. As an example, the space of symbolic strings on a finite number of symbols is proved to possess the property. Moreover, Sierpinski fractals, Koch curve as well as Cantor set satisfy the definition. A similarity map is introduced and the problem of chaos presence for the sets is solved by considering the dynamics of the map. This is true for Poincare, Li-Yorke and Devaney chaos, even in multi-dimensional cases. Original numerical simulations which illustrate the results are delivered.