Abstract
We consider a family of fractal squares, denoted as [Formula: see text]. Each of them satisfies the set equation [Formula: see text] for some [Formula: see text] with [Formula: see text]. It is known that two of these fractal squares are Lipschitz equivalent if and only if they are isometrically equivalent. The aim of our study is to improve this by replacing Lipschitz equivalence with topological equivalence. To this end, we shall investigate the group [Formula: see text] of all homeomorphisms of a fractal square [Formula: see text] that has a cut point and show that [Formula: see text] or [Formula: see text].
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