发表机构
Centro de Matemática, Computação e Cognição - UFABC; Instituto de Matemática e Estatística - USP; Instituto de Matemática, Estatística e Computação Científica - UNICAMP; Max-Planck-Institute for Mathematics in the Sciences（UFABC认知计算数学中心; 圣保罗大学数学与统计学院; 坎皮纳斯大学数学、统计与科学计算研究所; 马普科学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明闭黎曼流形$C^2$覆盖映射的前推作用在正$C^1$密度不变测度处Gâteaux可微，明确导数结构，并通过环面自同态的例子表明一阶刚性在任意维数下失效，揭示高维刚性现象的非线性本质。
AI 中文摘要
针对闭黎曼流形$M$的自同态$\phi$，我们研究了Wasserstein空间$\mathcal{P}(M)$上的前推作用$\phi_\ast$在$\phi$所保持的测度$\mu_0$处的性质。我们证明，若$\phi$是$C^2$覆盖映射且$\mu_0$具有正的$C^1$密度，则$\phi_\ast$在$\mu_0$处沿切方向是Gâteaux可微的，其导数由作用于向量场的$\phi$的转移算子，再经切空间上的正交投影给出。该导数是限制在切空间上的Koopman算子的伴随算子，其不动空间由$\mu_0$可在保持一阶不变性的前提下发生形变的方向构成。针对$\mathbb{T}^d$的适当自同态对，我们计算了它们的不动空间的交集，证明其包含一族无限的线性无关连续向量场，并对每个$n$构造了一个嵌入的$n$维测度族，这些测度在两个自同态作用下都近似不变。因此一阶刚性在任意维数下都不成立。特别是，诸如Furstenberg猜想的高维类似物这类刚性现象，若成立，则本质上是非线性的。
英文摘要
For an endomorphism $ϕ$ of a closed Riemannian manifold $M$, we study the pushforward action $ϕ_\ast$ on the Wasserstein space $\mathcal{P}(M)$ at a measure $μ_0$ preserved by $ϕ$. We show that if $ϕ$ is a $C^2$ covering map and $μ_0$ has positive $C^1$ density, then $ϕ_\ast$ is Gâteaux differentiable at $μ_0$ along tangent directions, with derivative given by the transfer operator of $ϕ$ acting on vector fields, followed by orthogonal projection onto the tangent space. The derivative is the adjoint of the Koopman operator restricted to the tangent space, and its fixed space consists of the directions in which $μ_0$ can be deformed while preserving invariance to first order. For appropriate pairs of endomorphisms of $\mathbb{T}^d$, we compute the intersection of their fixed spaces, show that it contains an infinite family of linearly independent continuous vector fields, and construct, for every $n$, an embedded $n$-dimensional family of measures that are nearly invariant under both endomorphisms. First-order rigidity therefore fails in every dimension. In particular, rigidity phenomena such as higher-dimensional analogues of the Furstenberg conjecture, if true, are genuinely nonlinear.
