Generic Zero-Entropy Optimization for Finitely Generated Nonlacunary Actions on the Circle
Abstract
Let Tn(x)=nxpmod 1 on T=ℝ/ℤ, and let Σ⊂ℕ be a finitely generated nonlacunary multiplicative semigroup. We prove that, for every 1≤ s≤∞, there is an open dense set of potentials f∈ W1,s(T) such that every measure that maximizes or minimizes ∫ f dμ over the Σ-invariant probability measures satisfies hμ(Tr)=0 for all r∈Σ∖{1}. For each fixed s, a single dense Gδ subset of W1,s(T) works simultaneously for all finitely generated nonlacunary multiplicative semigroups. The proof combines Rudolph--Johnson entropy rigidity with periodic-grid perturbations that generically exclude Haar measure from the optimizing faces. In particular, without any uniqueness assumption, the result applies to simultaneous Tp,Tq-invariance whenever p,q≥2 are multiplicatively independent.
Related
BibTeX
@misc{zhao2026generic,
title = {{Generic Zero-Entropy Optimization for Finitely Generated Nonlacunary Actions on the Circle}},
author = {Hang Zhao},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.24503},
eprint = {2608.24503},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24503},
}
RIS
TY - UNPB
TI - Generic Zero-Entropy Optimization for Finitely Generated Nonlacunary Actions on the Circle
AU - Hang Zhao
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.24503
UR - https://arxiv.org/abs/2608.24503
AB - Let T_n(x)=nxpmod 1 on T=ℝ/ℤ, and let Σ⊂ℕ be a finitely generated nonlacunary multiplicative semigroup. We prove that, for every 1≤ s≤∞, there is an open dense set of potentials f∈ W^1,s(T) such that every measure that maximizes or minimizes ∫ f dμ over the Σ-invariant probability measures satisfies h_μ(T_r)=0 for all r∈Σ∖{1}. For each fixed s, a single dense G_δ subset of W^1,s(T) works simultaneously for all finitely generated nonlacunary multiplicative semigroups. The proof combines Rudolph--Johnson entropy rigidity with periodic-grid perturbations that generically exclude Haar measure from the optimizing faces. In particular, without any uniqueness assumption, the result applies to simultaneous T_p,T_q-invariance whenever p,q≥2 are multiplicatively independent.
ER -