Quasisymmetrically co-Hopfian Sierpi\'nski Spaces and Menger Curve

TL;DR

This paper constructs new examples of complex fractal spaces, specifically Menger and Sierpiński spaces, that are quasisymmetrically co-Hopfian, and explores their classification and properties within geometric analysis.

Contribution

It provides the first examples of such spaces that are quasisymmetrically co-Hopfian and analyzes the uncountability of their quasisymmetric classes, extending prior work by Merenkov.

Findings

Constructed the first quasisymmetrically co-Hopfian Menger and Sierpiński spaces.

Proved the collection of quasisymmetric classes of Menger spaces is uncountable.

Generalized previous results by Merenkov on co-Hopfian properties.

Abstract

A metric space $X$ is quasisymmetrically co-Hopfian if every quasisymmetric embedding of $X$ into itself is onto. We construct the first examples of metric spaces homeomorphic to the universal Menger curve and higher dimensional Sierpi\'nski spaces, which are quasisymmetrically co-Hopfian. We also show that the collection of quasisymmetric equivalence classes of spaces homeomorphic to the Menger curve is uncountable. These results answer a problem and generalize results of Merenkov from \cite{Mer:coHopf}.