Finite-time Lyapunov diagnostics for nonstationary matrix products

Definitions and mathematical statement

The source asks what can be said about spectral properties and Lyapunov spectra in sequential or nonstationary settings. It does not specify a class of maps, operator spaces, cocycles, or a meaning of spectrum. One finite-dimensional interpretation in the attached aid is as follows. For invertible real $m\times m$ matrices $A_0,A_1,\ldots$, put \[ P_n=A_{n-1}\cdots A_0,\qquad \chi_j(n)=\frac1n\log s_j(P_n),\quad s_1(P_n)\ge\cdots\ge s_m(P_n)>0, \] using Euclidean singular values. The Lyapunov exponents in this interpretation are the limits of $\chi_j(n)$, when they exist. With induced Euclidean norms on exterior powers, set \[ \gamma_k(n)=\frac1n\log\|\bigwedge^kP_n\| =\sum_{j=1}^k\chi_j(n),\quad 1\le k\le m. \] The aid reports that convergence of all ordered singular exponents is equivalent to convergence of all these exterior-power growth rates; when the limits are finite, $\lim\chi_k=\lim\gamma_k-\lim\gamma_{k-1}$, with $\gamma_0=0$.

For an exact moving-frame reduction \[ A_k=B_{k+1}D_kB_k^{-1},\quad D_k=\operatorname{diag}(e^{a_{k,1}},\ldots,e^{a_{k,m}}),\quad B_k\in\operatorname{GL}_m({\mathbb{R}}), \] write $\bar a_{n,j}=n^{-1}\sum_{k=0}^{n-1}a_{k,j}$, and let $\bar a_{n,(j)}$ be their decreasing rearrangement. An explicit endpoint-frame error is \[ R_n=\log\max\{\|B_n\|\|B_0^{-1}\|,\ \|B_n^{-1}\|\|B_0\|\}. \] The supplied quantitative statement is \[ \max_{1\le j\le m}|\chi_j(n)-\bar a_{n,(j)}|\le\frac{R_n}{n}. \] In particular, if the frame and its inverse have subexponential norm growth, then $R_n/n\to0$; convergence questions reduce to the sorted coordinate averages. The exact reduction and subexponential distortion are hypotheses, not conclusions for arbitrary nonstationary products.

As a contrasting example the aid reports a sequence drawn from four fixed determinant-one matrices, each factor and inverse having norm at most $2$, for which \[ \liminf_{n\to\infty}\chi_1(n)=0,\qquad \limsup_{n\to\infty}\chi_1(n)=\log2. \] On subsequences realizing these two endpoints, the products have only eigenvalue $1$. Thus bounded factors and inverses need not ensure convergence, and eigenvalues of finite products need not detect their singular-value growth. The supplied summary does not include the four matrices, their dimension, or the sequence, so no explicit construction is asserted here. These finite-dimensional diagnostics constitute only the reported partial treatment of the broader question; no stationary ergodic assumptions or general operator spectral theorem are supplied.

Short English statement

Investigate spectra and Lyapunov growth without stationarity. The aid treats finite-dimensional products: exterior powers encode singular exponents, an exact diagonal moving frame gives a quantitative error bound, and a reported bounded-matrix example shows that exponents can fail to converge even when selected products have only eigenvalue one.

Sources

• AIM workshop Stochastic methods for non-equilibrium dynamical systems, Problem 18: https://aimath.org/pastworkshops/noneqdynsysproblems.pdf.
• Local registry data/unsolvedmath/problems.json, permanent ID 20001403, source code catalogue entry; canonical location aim-dynamical-systems-notes.json, notes[60].
• Attached machine-generated research attempt 1: partial result, run status unknown, internal rigor label "proved", low novelty confidence. It is not peer reviewed and requires independent verification before citation. No separate literature reference is supplied.