Manhattan property of geodesic paths on self-affine carpets


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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Corresponding author

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).

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

Mathematics Subject Classification

  • 28A80