arXiv subjects
Publications and source records attributed to Attila Gáspár.
We give a characterization of the existence of Lipschitz surjections between dust-like self-similar sets of the same dimension under the assumption that the similarity ratios are commensurable. Our results apply more generally to graph-directed fractals.
Kolmogorov asked the following question: can every bounded measurable set in the plane be mapped onto a polygon by a 1-Lipschitz map with arbitrarily small measure loss? The answer is negative in general, however, the case of compact sets is still open. We present an equivalent form of the question for compact sets. Furthermore, we give a positive answer to Kolmogorov's question for specific classes of sets, most importantly, for planar sets with tube-null boundary. In particular, we show that the Sierpiński carpet can be mapped into the union of finitely many line segments by a 1-Lipschitz map with arbitrarily small displacements, answering a question of Balka, Elekes and Máthé.