Uniform Dimension Theorem for SLE in Natural Time
Abstract
For 0<κ<8, we prove a uniform dimension transformation theorem for SLE in natural time: on a single probability-one event, Hausdorff and packing dimensions of every time set are multiplied by d=1+κ/8, and the same holds for lower and upper box dimensions of every bounded time set. The result applies to self-similar SLE with stationary increments, chordal SLE in the half-plane, and chordal SLE in bounded piecewise C1 Jordan domains. In the last case, the box-dimension conclusion holds under a positive-angle endpoint hypothesis. We also establish boundary-inclusive local Hölder continuity of every order below 1/d for chordal SLE in natural time and identify the exact local dimension d of the natural occupation measure on compact time intervals.
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BibTeX
@misc{zhan2026uniform,
title = {{Uniform Dimension Theorem for SLE in Natural Time}},
author = {Dapeng Zhan},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02493},
eprint = {2611.02493},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02493},
}
RIS
TY - UNPB
TI - Uniform Dimension Theorem for SLE in Natural Time
AU - Dapeng Zhan
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02493
UR - https://arxiv.org/abs/2611.02493
AB - For 0<κ<8, we prove a uniform dimension transformation theorem for SLE in natural time: on a single probability-one event, Hausdorff and packing dimensions of every time set are multiplied by d=1+κ/8, and the same holds for lower and upper box dimensions of every bounded time set. The result applies to self-similar SLE with stationary increments, chordal SLE in the half-plane, and chordal SLE in bounded piecewise C^1 Jordan domains. In the last case, the box-dimension conclusion holds under a positive-angle endpoint hypothesis. We also establish boundary-inclusive local Hölder continuity of every order below 1/d for chordal SLE in natural time and identify the exact local dimension d of the natural occupation measure on compact time intervals.
ER -