PreprintCC BY 4.0arXiv
Strong Borel-Cantelli laws and clusters of visits for random intermittent maps
Abstract
Near the neutral fixed point of an intermittent map, a single entrance into a small target can be followed by a long run of consecutive visits, so the number of visits need not follow its mean. We study targets [0,un] for iid random compositions of Liverani-Saussol-Vaienti maps with a bounded parameter law that is regularly varying at its minimum α∈(0,1); this includes a single map. For un=cn−gL(n), where c>0, 0<g≤1 and L is slowly varying, the visit count is almost surely asymptotic to its mean if and only if ∑n n−1(nun)−(1−α)/α<∞. Under the stationary law, the count divided by its mean converges to one in L1 exactly when nun→∞. Thus clusters of visits separate these two modes of convergence.
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BibTeX
@misc{wang2026strong,
title = {{Strong Borel-Cantelli laws and clusters of visits for random intermittent maps}},
author = {Qihang Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03010},
eprint = {2611.03010},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03010},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Strong Borel-Cantelli laws and clusters of visits for random intermittent maps
AU - Qihang Wang
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03010
UR - https://arxiv.org/abs/2611.03010
AB - Near the neutral fixed point of an intermittent map, a single entrance into a small target can be followed by a long run of consecutive visits, so the number of visits need not follow its mean. We study targets [0,u_n] for iid random compositions of Liverani-Saussol-Vaienti maps with a bounded parameter law that is regularly varying at its minimum α∈(0,1); this includes a single map. For u_n=cn^−gL(n), where c>0, 0<g≤1 and L is slowly varying, the visit count is almost surely asymptotic to its mean if and only if ∑_n n^−1(nu_n)^−(1−α)/α<∞. Under the stationary law, the count divided by its mean converges to one in L^1 exactly when nu_n→∞. Thus clusters of visits separate these two modes of convergence.
ER -