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Square Sierpiński carpets and Lattès maps

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Abstract

We prove that every quasisymmetric homeomorphism of a standard square Sierpiński carpet \(S_p\), \(p\ge 3\) odd, is an isometry. This strengthens and completes earlier work by the authors (Bonk and Merenkov in Ann Math (2) 177:591–643, 2013, Theorem 1.2). We also show that a similar conclusion holds for quasisymmetries of the double of \(S_p\) across the outer peripheral circle. Finally, as an application of the techniques developed in this paper, we prove that no standard square carpet \(S_p\) is quasisymmetrically equivalent to the Julia set of a postcritically-finite rational map.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, CA, 90095, USA

    Mario Bonk

  2. Department of Mathematics, City College of New York and CUNY Graduate Center, New York, NY, 10031, USA

    Sergei Merenkov

Authors
  1. Mario Bonk
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  2. Sergei Merenkov
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Correspondence to Sergei Merenkov.

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Mario Bonk was supported by NSF Grants DMS-1506099 and DMS-1808856. Sergei Merenkov was supported by NSF Grant DMS-1800180.

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Bonk, M., Merenkov, S. Square Sierpiński carpets and Lattès maps. Math. Z. 296, 695–718 (2020). https://doi.org/10.1007/s00209-019-02435-1

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