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Asymptotics for Functions Associated with Heat Flow on the Sierpinski Carpet

Published online by Cambridge University Press:  20 November 2018

B. M. Hambly
B. M. Hambly*
Affiliation:
Mathematical Institute, University of Oxford, Oxford, U.K. email: hambly@maths.ox.ac.uk
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Abstract

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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}^{\text{ }\!\!\gamma\!\!\text{ }}}\phi \left( \log x \right)$ for a suitable exponent $\text{ }\!\!\gamma\!\!\text{ }$ and $\phi $ a periodic function. We also discuss similar results for the heat content of affine nested fractals.

Keywords

35K0528A8035B4060J65
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 1 , 01 February 2011 , pp. 153 - 180
Copyright
Copyright © Canadian Mathematical Society 2011

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