Visibility of non-self-similar sets

TL;DR

This paper explores the visibility properties of non-self-similar fractal sets, specifically analyzing the structure of transformed Cantor sets and revealing conditions under which their visible sets contain intervals.

Contribution

It introduces a novel approach combining fractal theory, numerical methods, and dynamical systems to analyze non-self-similar sets' visibility, extending beyond previous self-similar focus.

Findings

The set $F^2_$ lacks self-similarity.

The visible set can contain a closed interval for certain .

New characterization method for non-self-similar sets' visibility.

Abstract

The visible problem is related to the arithmetic on the fractals. The visibility of self-similar set has been studied in the past. In this work, we investigate the visibility of non-self-similar sets. We begin by analyzing the structure of $F^2_\lambda$, where $F^2_{\lambda}:=\set{x^2:x\in F_{\lambda}}$ and $F_{\lambda}$ is the middle $1-2\lambda$ Cantor set, we show that it lacks self-similarity. Due to the nonlinear phenomena exhibited by $F^2_\lambda$, we develop a different approach to characterize the visible set. %combining methods from fractal theory, numerical computation, and dynamical systems theory. Our results also reveal that the visible set may contain a closed interval within a large range of $\lambda$.