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We investigate the relationship between algorithmic fractal dimensions and the classical local fractal dimensions of outer measures in Euclidean spaces. We introduce global and local optimality conditions for lower semicomputable outer measures. We prove that globally optimal outer measures exist. Our main theorem states that the classical local fractal dimensions of any locally optimal outer measure coincide exactly with the algorithmic fractal dimensions. Our proof uses an especially convenient locally optimal outer measureκdefined in terms of Kolmogorov complexity. We discuss implications for point-to-set principles.
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The strong maximum principle for Schrödinger operators on fractals
Abstract We prove a strong maximum principle for Schrödinger operators defined on a class of postcritically finite fractal sets and their blowups without boundary. Our primary interest is in weaker regularity conditions than have previously appeared in the literature; in particular we permit both the fractal Laplacian and the potential to be Radon measures on the fractal. As a consequence of our results, we establish a Harnack inequality for solutions of these operators.
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- Award ID(s):
- 1814253
- PAR ID:
- 10172798
- Date Published:
- Journal Name:
- Demonstratio Mathematica
- Volume:
- 52
- Issue:
- 1
- ISSN:
- 2391-4661
- Page Range / eLocation ID:
- 404 to 409
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1515/dema-2019-0034
