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From a Gap in the Lyapunov Spectrum to Dominated Splittings
Abstract
Let A be an α-Hölder cocycle over a transitive two-sided subshift of finite type, and assume that A is strongly bunched. We prove that if there exist k∈{1,…,d−1} and c>0 such that λk(p)−λk+1(p)≥ c for every periodic point p, then A admits a dominated splitting of index k. Thus, under strong bunching, a uniform asymptotic gap in the Lyapunov spectrum of periodic measures implies a uniform geometric splitting along all orbit segments. This extends the periodic-gap criterion of Kassel--Potrie from locally constant cocycles to Hölder cocycles in arbitrary dimension, and extends the two-dimensional fiber-bunched result of Velozo to higher dimensions under the stronger bunching assumption.
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BibTeX
@misc{backes2026from,
title = {{From a Gap in the Lyapunov Spectrum to Dominated Splittings}},
author = {Lucas Backes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28384},
eprint = {2609.28384},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28384},
}
RIS
TY - UNPB
TI - From a Gap in the Lyapunov Spectrum to Dominated Splittings
AU - Lucas Backes
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28384
UR - https://arxiv.org/abs/2609.28384
AB - Let A be an α-Hölder cocycle over a transitive two-sided subshift of finite type, and assume that A is strongly bunched. We prove that if there exist k∈{1,…,d−1} and c>0 such that λ_k(p)−λ_k+1(p)≥ c for every periodic point p, then A admits a dominated splitting of index k. Thus, under strong bunching, a uniform asymptotic gap in the Lyapunov spectrum of periodic measures implies a uniform geometric splitting along all orbit segments. This extends the periodic-gap criterion of Kassel--Potrie from locally constant cocycles to Hölder cocycles in arbitrary dimension, and extends the two-dimensional fiber-bunched result of Velozo to higher dimensions under the stronger bunching assumption.
ER -