A fundamental question raised by M. Kac in 1966 is: Must two isospectral planar domains necessarily be isometric? Following a long history of investigation C. Gordon, D. L. Webb and S. Wolpert in 1992 finally proved that the answer is no. By using the idea of transposition maps one can construct a wide class of planar domains with piecewise continuous boundaries which are isospectral but nonisometric. In this note we study the Kac question in relation to domains with fractal boundaries and by following a technique of paper folding maps by J. Chapman 1993 we construct pairs of isospectral but nonisometric connected planar domains with fractal boundaries having the same Minkowski dimension.
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Brian D. Sleeman, Hua Chen, On nonisometric isospectral connected fractal domains. Rev. Mat. Iberoam. 16 (2000), no. 2, pp. 351–361DOI 10.4171/RMI/278