5 The one-sided multiplicative ergodic theorem
This is the crux of the development. We work over a probability space \((X,\mu )\) with an ergodic measure-preserving transformation \(T\) and a measurable cocycle generator \(A : X \to \mathrm{Mat}_{d\times d}(\mathbb {R})\) with \(\det (A x)\neq 0\), subject to the one-sided integrability \(\log ^{+}\left\lVert A \right\rVert ,\, \log ^{+}\left\lVert A^{-1} \right\rVert \in L^1(\mu )\). The matrices act on \(\mathbb {R}^d\) (the Euclidean space) through the operator \(v\mapsto A v\), so that the relevant norm is the \(L^2\) operator norm, which is submultiplicative; we write \(A^{(n)}(x)\) for the cocycle \(A(T^{n-1}x)\cdots A(x)\). The whole chapter is organized around a single positive semidefinite limiting matrix, the Oseledets limit
whose eigenspaces, once their growth rates are matched to the deterministic singular-value exponents, yield the filtration of the target theorem.
5.1 The Oseledets limit
The candidate approximants are the symmetric positive roots \(q_n(x) := \bigl((A^{(n)}(x))^{\top } A^{(n)}(x)\bigr)^{1/2n}\), realized through the continuous functional calculus on the Gram matrix. The eigenvalues of \(q_n(x)\) are the \(1/n\)-th roots of the singular values of \(A^{(n)}(x)\); the heart of this section is that they converge a.e. to the exponentials of the deterministic Lyapunov exponents.
There is an antitone sequence \(\lambda ^0 : \mathbb {N}\to \mathbb {R}\) (antitone on \([0,d)\)) such that, for every \(i{\lt}d\) and \(\mu \)-a.e. \(x\),
where \(\sigma _i\) is the \(i\)-th sorted singular value.
Package the ergodic limits \(\Gamma _k = \lim \tfrac 1n \log s_k(A^{(n)}(x))\) of the products of the top \(k\) singular values, \(0\le k\le d\), obtained from Kingman’s subadditive ergodic theorem applied to the subadditive cocycle \(\log \left\lVert \textstyle \bigwedge ^{k}A^{(n)} \right\rVert \) (the exterior-power functor turns submultiplicativity of the \(\textstyle \bigwedge ^{k}\) operator norms into subadditivity). Then \(\lambda ^0_i := \Gamma _{i+1}-\Gamma _i\); the difference of the two a.e. \(\Gamma \)-limits gives the \(i\)-th singular-value exponent. Antitonicity of consecutive \(\lambda ^0_i\) descends from the antitone ordering of the singular values inside each \(n\), and chaining yields full antitonicity on \([0,d)\).
Let \(\mathrm{stepVal}\, \lambda ^0\, D\) be the step function \(e^{\lambda ^0_{D-1}} + \sum _{k=1}^{D-1}(e^{\lambda ^0_{k-1}}-e^{\lambda ^0_k})\, \mathbf1_{(c_k,\infty )}\) with thresholds \(c_k = e^{(\lambda ^0_k+\lambda ^0_{k-1})/2}\) strictly inside the \(k\)-th gap. If \(\lambda ^0\) is antitone on \([0,D)\) and \(j{\lt}D\), then \(\mathrm{stepVal}\, \lambda ^0\, D\, (e^{\lambda ^0_j}) = e^{\lambda ^0_j}\).
At the argument \(e^{\lambda ^0_j}\) the threshold indicator \(\mathbf1_{(c_k,\infty )}\) is \(1\) exactly when \(k{\gt}j\) (since \(\lambda ^0\) is antitone and \(c_k\) lies strictly between \(\lambda ^0_k\) and \(\lambda ^0_{k-1}\)), so only the increments above index \(j\) survive. Those increments \(e^{\lambda ^0_{k-1}}-e^{\lambda ^0_k}\) telescope to \(e^{\lambda ^0_j}-e^{\lambda ^0_{D-1}}\), which added to the constant base \(e^{\lambda ^0_{D-1}}\) returns \(e^{\lambda ^0_j}\).
For a self-adjoint matrix \(M\) and any function \(g\),
where \(\mu _j\) ranges over the sorted eigenvalues of \(M\).
Writing \(M = \mathrm{id}(M)\) gives \(M - g(M) = (\mathrm{id}-g)(M)\) by linearity of the continuous functional calculus, and the operator norm of a self-adjoint matrix functional calculus is the largest absolute eigenvalue \(\max _j\left\lvert \mu _j-g(\mu _j) \right\rvert \), which is bounded by the full nonnegative sum.
For \(\mu \)-a.e. \(x\) and every threshold index \(k\in [1,d)\), the block term \((e^{\lambda ^0_{k-1}}-e^{\lambda ^0_k})\cdot P^{c_k}_n(x)\) converges, where \(P^{c}_n\) is the band projector \(\mathbf1_{(c,\infty )}(q_n(x))\).
At a genuine gap \(\lambda ^0_k {\lt} \lambda ^0_{k-1}\) the threshold \(c_k\) is strictly separated from the two limiting eigenvalue clusters, so once the sorted eigenvalues of \(q_n(x)\) have converged the count of eigenvalues above \(c_k\) stabilizes and the corresponding spectral projector is Cauchy in operator norm. At a non-gap (\(\lambda ^0_{k-1}=\lambda ^0_k\)) the coefficient \(e^{\lambda ^0_{k-1}}-e^{\lambda ^0_k}\) vanishes, so the term is constantly \(0\).
For \(\mu \)-a.e. \(x\) the approximants \(q_n(x)\) converge in the matrix metric to a single matrix \(\Lambda (x)\).
The eigenvalues \(\mu _{j,n}=\sigma _j^{1/n}\) of \(q_n(x)\) converge a.e. to the exponentials \(e^{\lambda ^0_j}\) (Theorem 5.1). Form the block approximant \(\Lambda _n(x) := \mathrm{stepVal}\, \lambda ^0\, d\, (q_n(x))\), a finite linear combination of band projectors. By Lemma 5.3, \(\left\lVert q_n(x)-\Lambda _n(x) \right\rVert \le \sum _j\left\lvert \mu _{j,n}-\mathrm{stepVal}(\mu _{j,n}) \right\rvert \); each summand is eventually \(\left\lvert \mu _{j,n}-e^{\lambda ^0_j} \right\rvert \to 0\) because the step function reproduces the exponentials on the spectrum (Lemma 5.2). Meanwhile \(\Lambda _n(x)\) converges as a finite sum of convergent band-projector terms (Lemma 5.4). Adding the two convergences gives \(q_n(x)\to \Lambda (x)\), and the limit is selected pointwise.
\(\Lambda (x) := (\, \lim _n (q_n(x))_{ij}\, )_{ij}\) is the entrywise real \(\limsup \)/limit of the matrix entries of \(q_n(x)\); it is a total, measurable function of \(x\).
For \(\mu \)-a.e. \(x\), \(q_n(x)\to \Lambda (x)\) in the matrix metric, and \(\Lambda \) is measurable.
On the a.e. full convergence set of Theorem 5.5 the entrywise limit recovers the matrix limit (matrix convergence in finite dimensions is entrywise), so the entrywise \(\liminf \) defining \(\Lambda \) equals the genuine limit. Measurability is entrywise: each entry is a limit of measurable functions of \(x\), and a \(\liminf \) of measurable \(\mathbb {R}\)-valued functions is measurable.
For \(\mu \)-a.e. \(x\), \(\Lambda (x)\) is self-adjoint and positive semidefinite.
Self-adjointness \(M^{\top }=M\) is an entrywise closed condition preserved under the matrix limit of the self-adjoint approximants \(q_n(x)\). For positive semidefiniteness, the quadratic form \(M\mapsto v^{\top }Mv\) is continuous, so \(v^{\top }\Lambda (x)v = \lim _n v^{\top } q_n(x) v \ge 0\) as a limit of nonnegatives.
5.2 The per-vector lower bound
The lower half of the exact growth law isolates one band of the spectrum of \(q_n(x)\) and shows that a vector with nonzero projection onto the band grows at least at the band rate.
For self-adjoint \(Q\), a band indicator \(\chi =\mathbf1_{(c,\infty )}\), and a continuous \(f\ge 0\) on \(\mathrm{spec}(Q)\) with \(a\le f(t)\) whenever \(c{\lt}t\),
The band projector \(\chi (Q)\) is a self-adjoint idempotent, so \(\left\lVert \chi (Q)v \right\rVert ^2 = \langle \chi (Q)v,v\rangle \). The gap operator \((f-a\chi )(Q)\) is positive semidefinite because \(f-a\chi \ge 0\) on the spectrum (above \(c\), \(f\ge a=a\chi \); below, \(\chi =0\) and \(f\ge 0\)). Expanding \(\langle (f-a\chi )(Q)v,v\rangle \ge 0\) gives the claim.
For \(c\ge 0\) and \(n\ge 1\),
Raising \(q_n(x)=(\mathrm{gram}_n)^{1/2n}\) to the \(2n\)-th power via the functional calculus recovers the Gram matrix \(\mathrm{gram}_n=(A^{(n)})^{\top }A^{(n)}\) (the composed powers compose to the identity on the nonnegative spectrum). Apply Lemma 5.9 with \(f(t)=t^{2n}\) and \(a=c^{2n}\): above \(c\) one has \(t^{2n}\ge c^{2n}\). The right-hand inner product is then \(\langle \mathrm{gram}_n v,v\rangle = \left\lVert A^{(n)}(x)v \right\rVert ^2\).
If \(P^{c}_n(x)\to P\) with \(Pv\neq 0\), then \(\tfrac 1n\log \left\lVert P^{c}_n(x)\, v \right\rVert \to 0\).
The evaluation \(M\mapsto Mv\) is continuous in finite dimensions, so \(P^{c}_n(x)v\to Pv\neq 0\) and \(\left\lVert P^{c}_n(x)v \right\rVert \to \left\lVert Pv \right\rVert {\gt}0\). Hence the log converges to the finite number \(\log \left\lVert Pv \right\rVert \), and dividing by \(n\to \infty \) sends it to \(0\).
If \(P^{c}_n(x)\to P\) with \(c{\gt}0\) and \(Pv\neq 0\), and the cocycle growth sequence is cobounded, then
Taking logs in Lemma 5.10 and dividing by \(2n\) gives, eventually,
The left side converges to \(\log c\) since the band-correction term vanishes (Lemma 5.11). Passing to the \(\liminf \) along the inequality, using its boundedness and the coboundedness of the right side (supplied by the Furstenberg–Kesten integrability of the top exponent), yields the bound.
For thresholds \(c\le c'\) with limit band projectors \(P,P'\), if \(Pv=0\) then \(P'v=0\).
The finite-\(n\) bands are nested: \(\mathbf1_{(c,\infty )}\cdot \mathbf1_{(c',\infty )}= \mathbf1_{(c',\infty )}\) on the spectrum since \((c',\infty )\subseteq (c,\infty )\), so \(P^{c}_n P^{c'}_n = P^{c'}_n\); passing to the limit gives \(PP'=P'\). Both limit projectors are symmetric (limits of self-adjoint matrices), so transposing gives \(P'P=P'\), whence \(P'v = P'(Pv) = 0\).
5.3 The spectral upper bound and the determinant squeeze
The upper half is the genuinely non-elementary step. For a vector \(v\) in the slow subspace of \(\Lambda \) at level \(e^t\), one bounds \(\left\lVert A^{(n)}(x)v \right\rVert \) by the restricted operator norm of \(A^{(n)}\) on the slow subspace, whose growth exponent is pinned — non-circularly — by a determinant/volume squeeze.
If \(T\) is measure-preserving and \(g\ge 0\) with \(g\in L^1(\mu )\), then for \(\mu \)-a.e. \(x\), \(\tfrac 1n\, g(T^n x)\to 0\).
The series \(\sum _n g(T^n x)/n^{2}\) is a.e. finite by integrability and invariance of \(\mu \), so its terms \(g(T^n x)/n^2\to 0\); the Borel–Cantelli/sublinear-growth argument then gives \(g(T^n x)=o(n)\), i.e. \(\tfrac 1n g(T^n x)\to 0\).
Given a volume cocycle whose top, slow and remaining log-exponent sequences satisfy a sum law \(\mathrm{vol} = \mathrm{slow} + \mathrm{rest}\) with the appropriate limits, the slow restricted-operator-norm exponent obeys \(\limsup _n \tfrac 1n\log (\mathrm{slow}_n) \le \lambda _i\).
The total volume exponent is the Furstenberg–Kesten determinant limit \(\sum _j\lambda ^0_j\); the fast-block volume is the exterior-power Kingman limit; their difference forces the slow-block volume exponent. The squeeze converts a sum identity of limits into an upper bound for the slow factor once the fast and remaining factors are pinned, the angle/tilt between fast and slow blocks tempering to zero so no cross term inflates the slow volume.
Let \(v\neq 0\) lie in the slow subspace. Given the slow restricted-norm exponent bound \(\limsup _n \tfrac 1n\log r_n\le \lambda _i\) and the restriction estimate \(\left\lVert A^{(n)}(x)v \right\rVert \le r_n\left\lVert v \right\rVert \) (valid for \(v\) slow), one has
Take logs in \(\left\lVert A^{(n)}(x)v \right\rVert \le r_n\left\lVert v \right\rVert \), divide by \(n\), and pass to the \(\limsup \): the \(\left\lVert v \right\rVert \) factor contributes \(0\) and \(r_n\) contributes \(\limsup _n\tfrac 1n\log r_n\le \lambda _i\) (Lemma 5.15). The restricted-norm exponent \(r_n\) depends only on the global volume cocycle and the tempered slow–fast angle, never on the growth of \(v\) itself, so the argument is non-circular.
On the ultrametric-growth good set, every vector \(v\) of the \(\Lambda \)-slow band \(\mathrm{vslow}(e^t)\) with \(\limsup _n\tfrac 1n\log \left\lVert A^{(n)}v \right\rVert \le t\) lies in the growth sublevel \(\{ v:\overline\lambda (x,v)\le t\} \).
The \(\limsup \) of \(\tfrac 1n\log \left\lVert A^{(n)}v \right\rVert \) is by definition the upper growth function \(\overline\lambda (x,v)\); the hypothesis says it is \(\le t\), which is exactly membership in the sublevel \(\mathrm{lambdaSublevel}\, t\). The zero vector lies in every submodule.
5.4 Spectral identification of the filtration
The two bounds match the spectral filtration of \(\Lambda \) with the analytic limsup filtration. The bridge is that the finite-\(n\) band projectors converge a.e. to the functional calculus indicator of \(\Lambda \).
For \(\mu \)-a.e. \(x\), every \(c{\gt}0\) that is not one of the limiting eigenvalues \(e^{\lambda _{\mathrm{sing}}(x,i)}\) of \(\Lambda (x)\) satisfies
Since \(c\) avoids the spectrum of \(\Lambda (x)\), there is a gap \(\delta {\gt}0\) between \(c\) and every eigenvalue. Replace the discontinuous indicator by a continuous \(\delta /2\)-clamp surrogate \(\chi \) that is Lipschitz and agrees with \(\mathbf1_{(c,\infty )}\) at distance \(\ge \delta /2\) from \(c\). Once the sorted eigenvalues of \(q_n(x)\) are within \(\delta /2\) of their limits, \(\chi \) and \(\mathbf1_{(c,\infty )}\) agree on both spectra, so \(P^{c}_n(x)=\chi (q_n(x))\) eventually; by Lipschitz continuity of the functional calculus and \(q_n(x)\to \Lambda (x)\) (Theorem 5.7), \(\chi (q_n(x))\to \chi (\Lambda (x))=\mathbf1_{(c,\infty )}(\Lambda (x))\).
For \(\mu \)-a.e. \(x\) and every \(t\), \(\mathrm{lambdaSublevel}(x,t) \le \mathrm{vslow}(x,e^t)\).
Contrapositively, a vector \(v\notin \mathrm{vslow}(e^t)\) has nonzero component in the band of \(\Lambda (x)\) above \(e^t\); by Theorem 5.18 the finite band projectors converge to the corresponding CFC indicator with \(Pv\neq 0\), so the per-vector lower bound (Proposition 5.12) forces \(\liminf \tfrac 1n\log \left\lVert A^{(n)}v \right\rVert {\gt} t\), hence \(\overline\lambda (x,v){\gt}t\) and \(v\notin \mathrm{lambdaSublevel}\, t\). Kernel propagation across nested thresholds (Lemma 5.13) makes this consistent for all \(t\).
Under the spectral upper bound and the reverse inclusion, for \(\mu \)-a.e. \(x\) and every \(t\),
Two inclusions: the forward one (Theorem 5.17, from the upper bound) shows slow vectors grow slowly, hence lie in the sublevel; the reverse one (Theorem 5.19, from the lower bound) shows slowly-growing vectors lie in the slow space. Antisymmetry gives the identity, simultaneously for all \(t\) on the a.e. ultrametric-growth good set.
5.5 Ruelle’s reverse cofactor bound and the top-gap envelope
The upper bound on an individual vector reduces, after diagonalizing the limit, to controlling the overlap matrix between the sorted Gram eigenbasis at level \(n\) and the limiting eigenbasis. The graded overlap is controlled by a leakage induction; Ruelle’s cofactor estimate then converts a one-sided forward decay into the full pairwise rate.
Let \(S\) be orthogonal (\(S S^{\top }=1\)) with the graded forward decay \(\left\lvert S_{ab} \right\rvert \le c\cdot e^{-\max (g_b-g_a,0)}\). Then every entry obeys the reverse bound at the full pairwise rate:
Since \(S^{-1}=S^{\top }\), the entry \(S_{ij}\) is \((\det S)^{-1}\) times the cofactor \(\mathrm{adj}(S)_{ji}\), and \(\left\lvert \det S \right\rvert =1\) because \(SS^{\top }=1\). Expanding the minor by the Leibniz formula, every surviving permutation term collects the level imbalance \(g_i-g_j\) by telescoping the forward factors, each forward factor contributing at most \(c\, e^{-\max (\cdot ,0)}\); there are at most \((d-1)!\) such terms.
\(\mathrm{TopGapMassEnvelope}\, A\, T\, \lambda ^0\, x\) asserts the uniform geometric leakage of fast-band mass across each genuine gap: for every cut tolerance \(\delta \) there is a constant \(C\) controlling, eventually and uniformly, the band mass that crosses the top gap of each stratum.
If a nonnegative sequence \(a\) obeys a one-step recursion fed by finitely many source sequences each decaying geometrically with ratio \(\rho {\lt}1\), then \(a_n\) is bounded by a fixed multiple of the summed source envelopes for all \(n\).
Each single source contributes a geometric partial sum bounded by \(K/(1-\rho )\); summing the finitely many per-source envelopes and folding them through the linear one-step recursion gives a uniform bound on the chained quantity \(a_n\).
At a fixed cut strictly inside a gap of width \(\ge G\), the one-step band-mass increment is the current mass damped by \(e^{-G}\) plus a tempered source term; iterating produces the per-stratum leakage envelope.
In the sorted-Gram-eigenbasis block decomposition, the band mass above the cut at step \(n+1\) is the mass at step \(n\) attenuated by the singular-value ratio across the gap (bounded by \(e^{-nG}\)-type damping after \(n\) iterates) plus the contribution injected by the one-step generator \(A(T^n x)\), whose log-norm is tempered to \(o(n)\). This is exactly the geometric one-step recursion fed by tempered sources of Lemma 5.23.
For \(\mu \)-a.e. \(x\), the top-gap fast-band-mass envelope \(\mathrm{TopGapMassEnvelope}\, A\, T\, \lambda ^0\, x\) holds.
Fix the deterministic distinct gap \(G{\gt}0\) separating distinct exponents. On the a.e. set where every singular-value exponent converges (Theorem 5.1) and the one-step generator log-norm is tempered (Lemma 5.14), build the per-stratum leakage envelope for each gap pair (Lemma 5.24) and assemble them into the top-gap envelope: each stratum’s fast-band mass crossing its top gap is geometrically damped, uniformly in the cut, by a single per-pair constant maximized over the finitely many pairs.
5.6 Constancy of the spectrum
The deterministic exponent set is constant in \(x\) by construction, so once the per-point spectrum is identified with it, ergodic constancy is automatic.
If at \(x\) every realized exponent is a deterministic one and every deterministic exponent is attained, then \(\mathrm{lyapunovSpectrum}(x) = \mathrm{distinctExp}\, \lambda ^0\, d\).
Both directions are finite-set inclusions; antisymmetry of \(\subseteq \) gives the equality of the two finite subsets of \(\mathbb {R}\).
Given a.e. that every realized value of the upper growth function is a deterministic exponent (upper inclusion) and every deterministic exponent is attained (lower inclusion), for \(\mu \)-a.e. \(x\),
On the ultrametric-growth good set the two Finset inclusions are equivalent to native statements about the upper growth function \(\overline\lambda (x,\cdot )\): the upper inclusion is the spectral-upper-bound output (each stratum value is a deterministic exponent), the lower inclusion is attainment from the lower bound. Lemma 5.26 then gives the identity with the deterministic constant set; since that set does not depend on \(x\), the identification is \(T\)-invariant and the spectrum is a.e. constant — ergodicity adds nothing beyond the deterministic value already in hand.
5.7 Assembling the target theorem
The forward graded-overlap bound (built from the top-gap envelope and Ruelle’s reverse bound) gives the spectral upper bound \(\mathrm{hupper}\); combined with the reverse inclusion it yields the slow flag, and the lower bound supplies attainment. Together they discharge the three a.e. interfaces (spectrum, slow flag, exact growth) of the filtration assembly.
Assume the per-vector spectral upper bound on the slow flag, the reverse slow-flag inclusion, the two spectrum inclusions, and the band-projector convergence datum. Then there exist \(k\), strictly decreasing \(\lambda :\mathrm{Fin}\, k\to \mathbb {R}\), and a measurable family \(V\) forming a.e. a strictly decreasing \(A\)-equivariant flag along which \(\tfrac 1n\log \left\lVert A^{(n)}(x)v \right\rVert \to \lambda _i\) on each stratum.
The deterministic exponents \(\lambda ^0\) come from Theorem 5.1. The spectrum interface is discharged by the two inclusions (constancy, Theorem 5.27); the slow-flag interface by \(\mathrm{vslow}=\mathrm{lambdaSublevel}\) (Theorem 5.20); the exact-growth interface by combining the unconditional upper half (the stratum value \(\overline\lambda =\lambda _i\) on \(\mathrm{vflag}\)), the lower half (Proposition 5.12 fed the band datum), and Furstenberg–Kesten boundedness into a two-sided limit. These three interfaces feed the generic slow-flag assembly.
Under the standing ergodic, invertible, log-integrable hypotheses, and assuming the top-gap envelope \(\mathrm{TopGapMassEnvelope}\) quantified over \(\lambda ^0\), the full Oseledets filtration conclusion holds.
Diagonalize \(\Lambda (x)\) by its limit eigenbasis with eigenvalues \(e^{\lambda _{\mathrm{sing}}}\) and slow-orthogonality. The forward graded-overlap bound, consuming the envelope (Definition 5.22), yields the one-sided forward decay of the overlap matrix between the sorted Gram eigenbasis and the limit eigenbasis; Ruelle’s reverse cofactor estimate (Lemma 5.21) upgrades this to the full pairwise rate, which is exactly the slow-restriction bound feeding the spectral upper bound \(\mathrm{hupper}\) on the limit slow space. The band-projector convergence (Theorem 5.18) supplies the reverse slow-flag inclusion and the lower-bound datum, and the spectrum inclusions follow from the slow-flag identity. These discharge all hypotheses of Theorem 5.28.
Let \(\mu \) be a probability measure, \(T:X\to X\) ergodic measure-preserving, and \(A:X\to \mathrm{Mat}_{d\times d}(\mathbb {R})\) measurable with \(\det (A x)\neq 0\) and \(\log ^{+}\left\lVert A \right\rVert ,\, \log ^{+}\left\lVert A^{-1} \right\rVert \in L^1(\mu )\). Then there are \(k\) distinct Lyapunov exponents \(\lambda :\mathrm{Fin}\, k\to \mathbb {R}\), strictly decreasing, and a measurable family of subspaces
with each \(x\mapsto V_i\, x\) measurable, such that for \(\mu \)-a.e. \(x\): \(V_0\, x=\top \), \(V_k\, x=\bot \); the flag is strictly decreasing, \(V_{i+1}\, x {\lt} V_i\, x\); it is \(A\)-equivariant, \(A(x)\, V_i\, x = V_i\, (Tx)\); and along it the cocycle grows at the exact rate \(\lambda _i\):
If \(d=0\) the trivial flag \(\top =\bot \) with no exponents discharges the statement. For \(d{\gt}0\), the top-gap envelope holds a.e. (Theorem 5.25), so the conditional assembly Theorem 5.29 applies directly and produces the exponents, the measurable equivariant flag, and the exact per-stratum growth limits.