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# Asymptotics for Functions Associated with Heat Flow on the Sierpinski Carpet

Published:2010-11-06
Printed: Feb 2011
• B. M. Hambly,
Mathematical Institute, University of Oxford, Oxford, U.K.
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## Abstract

We establish the asymptotic behaviour of the partition function, the heat content, the integrated eigenvalue counting function, and, for certain points, the on-diagonal heat kernel of generalized Sierpinski carpets. For all these functions the leading term is of the form $x^{\gamma}\phi(\log x)$ for a suitable exponent $\gamma$ and $\phi$ a periodic function. We also discuss similar results for the heat content of affine nested fractals.
 MSC Classifications: 35K05 - Heat equation 28A80 - Fractals [See also 37Fxx] 35B40 - Asymptotic behavior of solutions 60J65 - Brownian motion [See also 58J65]