#### Presentation Title

Spectral Dimension of a Class of One-Dimensional Fractal Laplacians

#### Document Type

Presentation

#### Presentation Date

6-13-2017

#### Abstract or Description

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 "bounded measure 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.

#### Sponsorship/Conference/Institution

Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals

#### Location

Ithaca, NY

#### Recommended Citation

Ngai, Sze-Man.
2017.
"Spectral Dimension of a Class of One-Dimensional Fractal Laplacians."
*Department of Mathematical Sciences Faculty Presentations*.
Presentation 370.

https://digitalcommons.georgiasouthern.edu/math-sci-facpres/370