On critical values of self-similar sets

TL;DR

This paper investigates the critical points of self-similar sets, establishing conditions for countability of critical values and constructing examples with uncountably many such values.

Contribution

It introduces a simple condition ensuring countably many critical values and constructs a self-similar set with uncountably many critical values, advancing understanding of their structure.

Findings

A simple condition implies at most countably many critical values.

Constructed a self-similar set with uncountably many critical values.

Enhanced understanding of the distribution of critical points in self-similar sets.

Abstract

In the paper we study properties of the set of critical points for self-similar sets. We introduce simple condition that implies at most countably many critical values and we construct a self-similar set with uncountable set of critical values.