圆周上有限生成非间隙作用的通用零熵优化
Generic Zero-Entropy Optimization for Finitely Generated Nonlacunary Actions on the Circle
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中文总结 AI 辅助
针对圆周上有限生成非间隙乘法半群作用，证明利普希茨空间中开稠密子集的不变极值测度熵为零，且存在稠密G_δ子集使该结论对所有此类半群同时成立，结合了周期网格扰动估计与Rudolph–Johnson定理。
中文摘要 AI 辅助
设Σ⊂ℕ是作用在环面𝕋=ℝ/ℤ上的有限生成非间隙乘法半群，作用方式为Tₙ(x)=nx mod 1。我们证明，对于利普希茨空间Lip(𝕋)中的一个开且稠密的子集，每个Σ-不变的极大或极小测度μ，都满足对所有n∈Σ∖{1}有h_μ(Tₙ)=0。此外，存在Lip(𝕋)的一个单一稠密G_δ子集，使得对于所有有限生成非间隙乘法半群Σ⊂ℕ，该结论同时成立。证明结合了周期网格扰动估计与Rudolph–Johnson正熵刚性定理。
英文摘要
Let $T_n(x)=nx\pmod 1$ on $\T=\mathbb R/\mathbb Z$, and let $Σ\subset\mathbb N$ be a finitely generated nonlacunary multiplicative semigroup. We prove that, for every $1\le s\le\infty$, there is an open dense set of potentials $f\in W^{1,s}(\T)$ such that every measure that maximizes or minimizes $\int f\,dμ$ over the $Σ$-invariant probability measures satisfies $h_μ(T_r)=0$ for all $r\inΣ\setminus\{1\}$. 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 $p,q\ge2$ are multiplicatively independent.
发表机构
- School of Mathematics and Information Science, Neijiang Normal University（内江师范学院数学与信息科学学院）
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