Manhattan property of geodesic paths on self-affine carpets

Abstract

For any Bedford-McMullen self-affine carpet, the geodesic path on the carpet between points \((x_{1},y_{1})\) and \((x_{2},y_{2})\) has length greater than or equal to \(|x_{1}-x_{2}|+|y_{1}-y_{2}|.\) This property fails for self-similar carpets.

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Correspondence to Lifeng Xi.

Additional information

The work is supported by National Natural Science Foundation of China (Nos. 11771226, 11371329, 11471124) and K.C. Wong Magna Fund in Ningbo University.

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Li, Y., Xi, L. Manhattan property of geodesic paths on self-affine carpets. Arch. Math. 111, 279–285 (2018). https://doi.org/10.1007/s00013-018-1199-4

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Keywords

  • Fractal
  • Self-affine carpet
  • Rectifiable curve
  • Manhattan distance

Mathematics Subject Classification

  • 28A80