修正的Lyons–Sidorova猜想的共振傅里叶树分解
Entire Logarithmic Signatures of Bounded-Variation Paths in Finite Dimensions
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中文总结 AI 辅助
该论文在有限维实赋范向量空间中证明修正的Lyons–Sidorova猜想，结合精确矩阵等谱性等方法，推导有界变差路径的特征与对数特征关系，实现路径级约化。
中文摘要 AI 辅助
设γ为有限维实赋范向量空间中连续有界变差路径，其特征为g=S(γ)，对数特征为l=log g，增量为v=γ_T−γ_0。我们在此情形下证明修正的Lyons–Sidorova猜想：若R(l)=∞，则当v=0时g=1；当v≠0时，中心化路径的实际前缀α满足S(γ)=S(α)e^v S(α)⁻¹。反之，所有此类有界变差特征均具有全对数特征。对于树约化路径，不同的规范前缀给出等价的弱路径共轭至线段。证明结合了精确矩阵等谱性、环刚性与共振傅里叶展开：Le Donne–Züst特征树中的不动点几何产生公共有理频率前缀，傅里叶唯一性重构普通特征，不变轴几何实现路径级约化。
英文摘要
We classify signatures of bounded-variation paths in finite-dimensional real normed spaces whose logarithms are entire, in the sense of superexponential homogeneous decay. The zero-first-level fibre is trivial; if the first level is $v\neq0$, the signature is $a\e^v a^{-1}$ with $a$ a bounded-variation signature, and every such conjugate is entire. For a given path, $a$ may be chosen from a prefix. For a tree-reduced representative, a gate-selected prefix gives weak path conjugacy to a line. The proof uses a finite-dimensional spectral-growth statement: if $q\geq1$, $A:\C\to M_q(\C)$ is entire, and $\norm{\e^{A(z)}}\leq C\e^{τ\abs{z}}$, then the $k$th characteristic-polynomial coefficient of $A(z)$ has degree at most $k$ for $1\leq k\leq q$. Universal matrix isospectrality, resonant developments, metric-tree fixed points and compactness, and Stieltjes--Fourier reconstruction establish the bounded-variation modified Lyons--Sidorova conjecture under a hypothesis imposed only on the whole path.
