On the topological dynamical system of tilings
Abstract
The compactness of the tiling space and the continuity of the action of the Euclidean space by translation, as well as the minimality of this action for repetitive tilings, are among the most frequently mentioned foundational facts in the dynamical theory of aperiodic order. Nonetheless, despite their classical status, a comprehensive treatise with complete and detailed proofs, to the best of our knowledge, is not available in the literature: existing arguments are typically sketched, deferred to other sources, or rely on unstated results or hypotheses. The purpose of this manuscript is to give a self-contained and pedagogical treatment of these foundations, providing complete and carefully explained proofs. We hope that this exposition will provide a useful reference and contribute to a clearer and more accessible understanding of the fundamentals of the theory of tiling dynamical systems.
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BibTeX
@misc{santander2026topological,
title = {{On the topological dynamical system of tilings}},
author = {José Pablo Santander},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06310},
eprint = {2609.06310},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06310},
}
RIS
TY - UNPB
TI - On the topological dynamical system of tilings
AU - José Pablo Santander
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.06310
UR - https://arxiv.org/abs/2609.06310
AB - The compactness of the tiling space and the continuity of the action of the Euclidean space by translation, as well as the minimality of this action for repetitive tilings, are among the most frequently mentioned foundational facts in the dynamical theory of aperiodic order. Nonetheless, despite their classical status, a comprehensive treatise with complete and detailed proofs, to the best of our knowledge, is not available in the literature: existing arguments are typically sketched, deferred to other sources, or rely on unstated results or hypotheses. The purpose of this manuscript is to give a self-contained and pedagogical treatment of these foundations, providing complete and carefully explained proofs. We hope that this exposition will provide a useful reference and contribute to a clearer and more accessible understanding of the fundamentals of the theory of tiling dynamical systems.
ER -