Abstract
The entropy of a plane curve is defined in terms of the number of intersection points with a random line. The Gibbs distribution which maximizes the entropy enables one to define the temperature of the curve. At 0 temperature, the curve reduces to a straight segment. At high temperature, the curve is somewhat chaotic and “behaves like a perfect gas”. We attempt to show that thermodynamic formalism can be used for the study of plane curves. The curves we discuss have finite length, unlike Mandelbrot's fractal curves [1], yet we feel our approach to the mathematics is not far from his.
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References
B. B. Mandelbrot, The fractal geometry of Nature, Freeman, 1982.
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Communicated by J. Serrin
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Dupain, Y., Kamae, T. & Mendés, M. Can one measure the temperature of a curve?. Arch. Rational Mech. Anal. 94, 155–163 (1986). https://doi.org/10.1007/BF00280431
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DOI: https://doi.org/10.1007/BF00280431