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A notion of Euler characteristic for fractals
Author(s) -
Llorente Marta,
Winter Steffen
Publication year - 2007
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200410471
Subject(s) - mathematics , sierpinski triangle , euler's formula , fractal , exponent , infinitesimal , euler number (physics) , euler characteristic , pure mathematics , proof of the euler product formula for the riemann zeta function , mathematical analysis , semi implicit euler method , euler equations , backward euler method , philosophy , linguistics , prime zeta function , arithmetic zeta function , riemann hypothesis
Abstract A notion of (average) fractal Euler number for subsets of ℝ d with infinite singular complexes is introduced by means of rescaled Euler numbers of infinitesimal ε ‐neighbourhoods. For certain classes of self‐similar sets we calculate the associated Euler exponent and the (average) fractal Euler number with the help of the Renewal theorem. Examples like the Sierpinski gasket or carpet are provided. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)