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A combinatorial integration on the Cantor dust

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arxiv 2209.03537 v2 pith:GPEHPVRH submitted 2022-09-08 math.KT math.MG

A combinatorial integration on the Cantor dust

classification math.KT math.MG
keywords cantorcocycledustcombinatorialdimensionalfunctionfunctionsintegration
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In this paper, we generalize the Cantor function to $2$-dimensional cubes and construct a cyclic $2$-cocycle on the Cantor dust. This cocycle is non-trivial on the pullback of the smooth functions on the $2$-dimensional torus with the generalized Cantor function while it vanishes on the Lipschitz functions on the Cantor dust. The cocycle is calculated through the integration of $2$-forms on the torus by using a combinatorial Fredholm module.

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