Spectral Asymptotics of Some One-Dimensional Fractal Laplacians

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The spectral dimension of the Laplacian defined by a measure has been shown to be closely related to heat kernel estimates, which under suitable conditions determine whether wave propagates with finite or infinite speed. We observe that some self-similar measures defined by finite or infinite iterated function systems with overlaps satisfy certain “essentially finite type condition”, which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain, under this condition, a closed formula for the spectral dimension of the Laplacian. Earlier results for fractal measures with overlaps rely on Strichartz second-order identities, which are not satisfied by the measures we consider here. This is a joint work with Wei Tang and Yuanyuan Xie.


Southeastern-Atlantic Regional Conference on Differential Equations (SEARCDE)


Kennesaw, GA