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Measure rigidity for regular polynomial endomorphisms and applications
Abstract
We classify regular polynomial endomorphisms of ℂk with the same equilibrium measure under some mild condition on the dynamics at infinity. In particular, for f in a Zariski dense open subset of the parameter space and any g of the same degree, μg=μf if and only if f=g. We also establish finiteness results for maps with a prescribed equilibrium measure. Our approach rests on the stable manifold structure near the hyperplane at infinity. Applications include a Tits-type alternative, results on iterated centralizers, and an arithmetic non-density theorem.
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BibTeX
@misc{zhang2026measure,
title = {{Measure rigidity for regular polynomial endomorphisms and applications}},
author = {Yugang Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.23749},
eprint = {2608.23749},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23749},
}
RIS
TY - UNPB
TI - Measure rigidity for regular polynomial endomorphisms and applications
AU - Yugang Zhang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2608.23749
UR - https://arxiv.org/abs/2608.23749
AB - We classify regular polynomial endomorphisms of ℂ^k with the same equilibrium measure under some mild condition on the dynamics at infinity. In particular, for f in a Zariski dense open subset of the parameter space and any g of the same degree, μ_g=μ_f if and only if f=g. We also establish finiteness results for maps with a prescribed equilibrium measure. Our approach rests on the stable manifold structure near the hyperplane at infinity. Applications include a Tits-type alternative, results on iterated centralizers, and an arithmetic non-density theorem.
ER -