On the Existence of a Self-Similar Coarse Graining of a Self-Similar Space

TL;DR

The paper proves that certain self-similar topological spaces maintain their self-similarity through coarse graining, with such spaces generated from intense quadratic dynamics, revealing an endless self-similar hierarchy.

Contribution

It demonstrates the existence of a self-similar space whose coarse graining is also self-similar, extending the understanding of self-similarity in topological spaces.

Findings

Self-similar spaces can be preserved under coarse graining.

Such spaces can be generated from intense quadratic dynamics.

The self-similarity property persists infinitely through successive coarse grainings.

Abstract

A topological space homeomorphic to a self-similar space is demonstrated to be self-similar. There exists a self-similar space $S$ whose coarse graining is homeomorphic to $S$. The coarse graining of $S$ is, therefore, self-similar again. In the same way, the coarse graining of the self-similar coarse graining of $S$ is, furthermore, self-similar. These situations succeed endlessly. Such a self-similar $S$ is generated actually from an intense quadratic dynamics.