AI 中文总结
该研究针对局部同胚的纤维收缩扩张问题，构建了上同调、变分与热力学问题的约化转移框架，证明了相关测度、熵等性质的对应关系，并通过仿射和符号收缩斜积示例验证了框架的有效性。
AI 中文摘要
我们为局部同胚的纤维收缩扩张中的上同调、变分和热力学问题构建了约化与转移框架。在纤维方向存在一致收缩且存在连续全局截面的条件下，每个Hölder势都可显式分解为φ=ψ∘π+u−u∘F，其中ψ定义在商空间上，u由一致收敛的纤维方向级数得到。我们给出了连续性和Hölder正则性的定量判据，并利用该约化将子上同调和Livšic型命题从商空间转移到扩张空间。我们证明了扩张空间与商空间上的不变测度存在双射、不变平均值保持不变，且纤维具有零相对拓扑熵。因此，度量熵、遍历优化、压强、极小测度和平衡态相互对应。我们进一步在Bowen球质量的次指数比较下建立了弱Gibbs转移，并沿指定的依赖点的Gibbs时间证明了分块Bowen序列弱Gibbs转移；在一致有界的序列质量畸变下，强Bowen序列Gibbs性质得以保持。我们还给出了Bowen球可比较性的具体判据。最后，仿射和符号收缩斜积对该框架进行了示例说明，包括针对F(x,y)=(f(x),ay+ρ(x))（|a|<1）的显式约化：φ(x,y)=g(x)+by，ψ(x)=g(x)+(b/(1−a))ρ(x)。
英文摘要
We develop a reduction and transfer framework for cohomological, variational, and thermodynamic problems in fiber-contracting extensions of local homeomorphisms. Under uniform contraction along the fibers and the existence of a continuous global section, every Hölder potential admits the explicit decomposition \[ φ=ψ\circπ+u-u\circ F, \] where $ψ$ is defined on the quotient and $u$ is obtained by a uniformly convergent fiberwise series. We give quantitative criteria for continuity and Hölder regularity and use the reduction to transfer subcohomological and Livšic-type statements from the quotient to the extension. We prove that invariant measures on the extension and quotient are in bijection, invariant averages are preserved, and the fibers have zero relative topological entropy. Consequently, metric entropy, ergodic optimization, pressure, minimizing measures, and equilibrium states correspond. We further establish weak Gibbs transfer under subexponential comparison of Bowen-ball masses and prove blockwise Bowen sequential weak Gibbs transfer along prescribed point-dependent Gibbs times; under uniformly bounded sequential mass distortion, the strong Bowen sequential Gibbs property is preserved. Concrete criteria for Bowen-ball comparability are provided. Finally, affine and symbolic contracting skew-products illustrate the framework, including the explicit reduction \[ φ(x,y)=g(x)+by \quad\;\;\;\;\;\;\;\;\quad ψ(x)=g(x)+\frac{b}{1-a}ρ(x) \] for $F(x,y)=(f(x),ay+ρ(x))$, $|a|<1$.
