PreprintCC BY 4.0arXiv
On the horizon problem over the product of self-similar fractals
Abstract
In this article, we first investigate the box dimension results for the graphs of continuous functions defined on F× F, where F is the self-similar fractal generated by an iterated function system (IFS) exhibiting the open set condition (OSC). Following this, we manifest the existence of a prevalent surface, defined by a continuous function on F× F, that satisfies the horizon property in relation to the box dimension, i.e., the box dimension of the prevalent surface is greater than that of its horizon up to a constant η, where η is the box dimension of the self-similar fractal F. Finally, we provide an example consolidating that the OSC on the underlying IFS of F cannot be dropped.
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BibTeX
@misc{gurubachan2026horizon,
title = {{On the horizon problem over the product of self-similar fractals}},
author = {Gurubachan and S. Verma and V. V. M. S. Chandramouli},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.21047},
eprint = {2608.21047},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.21047},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the horizon problem over the product of self-similar fractals
AU - Gurubachan
AU - S. Verma
AU - V. V. M. S. Chandramouli
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2608.21047
UR - https://arxiv.org/abs/2608.21047
AB - In this article, we first investigate the box dimension results for the graphs of continuous functions defined on F× F, where F is the self-similar fractal generated by an iterated function system (IFS) exhibiting the open set condition (OSC). Following this, we manifest the existence of a prevalent surface, defined by a continuous function on F× F, that satisfies the horizon property in relation to the box dimension, i.e., the box dimension of the prevalent surface is greater than that of its horizon up to a constant η, where η is the box dimension of the self-similar fractal F. Finally, we provide an example consolidating that the OSC on the underlying IFS of F cannot be dropped.
ER -