8 The continuous-flow multiplicative ergodic theorem
8.1 Overview
The discrete multiplicative ergodic theorem governs a single measure-preserving map \( T : X \to X \) and the iterates of a matrix cocycle generated one step at a time. This chapter lifts that theorem to continuous time: the map \( T \) is replaced by a one-parameter measure-preserving flow \( \varphi : \mathbb {R}\to X \to X \), and the iterated cocycle by a continuous-time linear cocycle \( A : \mathbb {R}\to X \to \mathrm{Mat}_d(\mathbb {R}) \). The conclusion is a finite Lyapunov spectrum \( \lambda _1 {\gt} \cdots {\gt} \lambda _k \), a measurable filtration that is flow-equivariant at every real time, and the exact continuous-parameter growth \( t^{-1} \log \left\lVert A(t,x)v \right\rVert \to \lambda _i \) as \( t \to \infty \) over \( \mathbb {R}\).
The strategy is a reduction, not a redevelopment of the ergodic machinery for \( \mathbb {R}\). We set \( T := \varphi (1) \) and read the discrete cocycle off the sampled flow cocycle; the proved discrete theorem 5.30 delivers the integer-time conclusion, and two analytic devices lift it to the continuous parameter: a between-times sandwich that controls growth on each interval \( [n, n+1) \), and a shift-invariance of the growth \( \limsup \) that promotes the discrete (time-one) equivariance to equivariance at every real time. No continuous-time Kingman theorem is needed; the integer clock appears only as a technical reduction device. Throughout, \( X \) carries no topology: the flow is a measurable measure-preserving action of \( (\mathbb {R}, +) \), and the cocycle is measurable in the state at each fixed time.
8.2 The continuous-time data
A measure-preserving one-parameter flow on a measurable space \( X \) for a measure \( \mu \) is a family \( \varphi : \mathbb {R}\to X \to X \) together with the data \( \varphi (0) = \mathrm{id} \), \( \varphi (s+t) = \varphi (s) \circ \varphi (t) \) for all \( s, t \in \mathbb {R}\), and a proof that each time-\( t \) map \( \varphi (t) \) preserves \( \mu \). No topology on \( X \) is assumed; in particular no joint continuity in \( (t, x) \) is required.
For a measure-preserving flow \( \varphi \) and \( n \in \mathbb {N}\), the integer-time map of the flow is the \( n \)-fold iterate of its time-one map: \( \varphi (n) = (\varphi (1))^{[n]} \).
Induction on \( n \). The base case \( \varphi (0) = \mathrm{id} = (\varphi (1))^{[0]} \) is the time-zero law. For the step, write \( n+1 = n + 1 \) as reals and apply additivity \( \varphi (n+1) = \varphi (n) \circ \varphi (1) \), then the inductive hypothesis and \( (\varphi (1))^{[n+1]} = (\varphi (1))^{[n]} \circ \varphi (1) \).
A continuous-time linear cocycle over a measure-preserving flow \( \varphi \), valued in invertible \( d \times d \) real matrices, is a family \( A : \mathbb {R}\to X \to \mathrm{Mat}_d(\mathbb {R}) \) with \( A(0,x) = 1 \), the cocycle identity (newest factor on the left) \( A(t+s,x) = A(t, \varphi (s)x)\, A(s,x) \), a proof that \( \det A(t,x) \ne 0 \) for all \( t, x \), and measurability of each time-\( t \) map \( A(t,\cdot ) \).
For a flow cocycle \( A \) over \( \varphi \), every \( n \in \mathbb {N}\) and every \( x \),
i.e. at integer times the continuous-time cocycle equals the discrete iterated cocycle generated by its time-one map \( A(1,\cdot ) \) over the time-one dynamics \( \varphi (1) \).
Induction on \( n \). At \( n = 0 \) both sides are the identity. For the step, split \( A((n+1),x) = A(n, \varphi (1)x)\, A(1,x) \) by the cocycle identity at \( t = n \), \( s = 1 \), match the recursion \( \mathrm{cocycle}\, (n+1)\, x = \mathrm{cocycle}\, n\, (\varphi (1)x) \cdot A(1,x) \), and apply the inductive hypothesis at the point \( \varphi (1)x \).
8.3 Reduction to the discrete theorem
The discrete theorem requires the time-one generator to have integrable positive log-norm, both forward and inverse. These follow by evaluating the uniform dominating hypotheses at \( s = 1 \).
If \( g \in L^1(\mu ) \) dominates \( \log ^{+}\left\lVert A(s,x) \right\rVert \) for all \( s \in [0,1] \) and all \( x \), then the time-one map \( A(1,\cdot ) \) has integrable positive log-norm.
The map \( x \mapsto \log ^{+}\left\lVert A(1,x) \right\rVert \) is measurable (composition of the measurable \( \log ^{+} \), the operator norm, and the measurable time-one cocycle map) and is dominated pointwise by \( g \), taking \( s = 1 \in [0,1] \) in the hypothesis. Dominated by an integrable function, it is integrable.
If \( g' \in L^1(\mu ) \) dominates \( \log ^{+}\left\lVert (A(s,x))^{-1} \right\rVert \) for all \( s \in [0,1] \) and all \( x \), then the inverse \( (A(1,\cdot ))^{-1} \) has integrable positive log-norm.
Identical to 8.5, inserting the measurable matrix-inversion map and evaluating the dominating hypothesis at \( s = 1 \).
Let \( \mu \) be a probability measure, \( \varphi \) a measure-preserving flow with \( \varphi (1) \) ergodic, and \( A \) a flow cocycle with the two uniform integrable dominators \( g, g' \). Then there exist \( k \), exponents \( \lambda : \mathrm{Fin}\, k \to \mathbb {R}\), and a subspace family \( V \) forming an Oseledets filtration for the generator \( A(1,\cdot ) \) over the dynamics \( \varphi (1) \).
8.4 Between integer times
The discrete theorem controls growth only along integer times; the next results bridge the gap to a continuous parameter. Both rest on the orbital sublinearity of an integrable function along the flow’s integer orbit.
For integrable \( g, g' \) and a measure-preserving flow \( \varphi \), for almost every \( x \) the combined fluctuation along the integer orbit vanishes:
Apply the Birkhoff orbital-tail estimate (a.e. \( n^{-1} h(T^{[n]}x) \to 0 \) for integrable \( h \)) to \( h = g + g' \) and the time-one map \( T = \varphi (1) \), then rewrite the iterate orbit \( (\varphi (1))^{[n]}x \) as \( \varphi (n)x \) using 8.2.
Fix a flow \( \varphi \), a flow cocycle \( A \) with uniform forward/inverse one-step controls \( g, g' \) on \( [0,1] \), a point \( x \), and a nonzero vector \( v \). Suppose the integer fluctuation error vanishes and the integer-time average converges, \( n^{-1}\log \left\lVert A(n,x)v \right\rVert \to L \). Then the continuous-time average converges to the same limit:
Write \( t = r + n \) with \( n = \lfloor t \rfloor \ge 1 \) and \( r \in [0,1) \). The cocycle identity splits \( A(t,x) = A(r, \varphi (n)x)\, A(n,x) \), so with \( w = A(n,x)v \) one sandwiches \( \log \left\lVert A(t,x)v \right\rVert \) between \( \log \left\lVert w \right\rVert - \log \left\lVert (A(r,\varphi (n)x))^{-1} \right\rVert \) and \( \log \left\lVert w \right\rVert + \log \left\lVert A(r,\varphi (n)x) \right\rVert \). Both correction terms are bounded in absolute value by \( g(\varphi (n)x) + g'(\varphi (n)x) \) via the \( [0,1] \) controls (using \( \left\lVert M \right\rVert \left\lVert M^{-1} \right\rVert \ge 1 \)). Dividing by \( t \), the error term vanishes by hypothesis, the discrete average converges to \( L \) along \( \lfloor t \rfloor \), and the floor ratio \( \lfloor t \rfloor / t \to 1 \); a squeeze over the floor delivers the continuous-time limit.
8.5 Equivariance at every real time
The discrete theorem gives equivariance one integer step at a time. To obtain equivariance at every real time \( t_0 \) we use the intrinsic growth characterization: a vector lies in level \( V_i \) iff it is zero or its discrete growth \( \limsup \) is \( \le \lambda _i \). The fixed matrix \( A(t_0,x) \) is a bounded bijection, so it perturbs the per-step log-norm by \( o(n) \), hence leaves the growth \( \limsup \) unchanged.
Fix a real time \( t_0 \). For almost every \( x \), both \( n^{-1}\log \left\lVert A(t_0, \varphi (n)x) \right\rVert \) and \( n^{-1}\log \left\lVert (A(t_0, \varphi (n)x))^{-1} \right\rVert \) tend to \( 0 \) as \( n \to \infty \).
One builds an integrable function \( H \) dominating both \( \log ^{+}\left\lVert A(t_0,\cdot ) \right\rVert \) and \( \log ^{+}\left\lVert (A(t_0,\cdot ))^{-1} \right\rVert \): by induction on \( \lfloor t_0 \rfloor \) one splits \( A((\rho +n)+1,\cdot ) = A(\rho +n,\varphi (1)\cdot )\, A(1,\cdot ) \), bounds \( \log ^{+} \) of a product by the sum, and uses measure-preservation of \( \varphi (1) \); negative times reduce to positive ones via \( A(t_0,y) = (A(-t_0,\varphi (t_0)y))^{-1} \). The two-sided bound \( \left\lvert \log \left\lVert M \right\rVert \right\rvert \le \log ^{+}\left\lVert M \right\rVert + \log ^{+}\left\lVert M^{-1} \right\rVert \) turns \( H \) into an integrable dominator for the absolute log-norm; the Birkhoff orbital tail of \( H \) along the integer orbit and a squeeze finish the proof.
Fix a real time \( t_0 \). For almost every \( x \) and every test vector \( u \), the discrete-time growth \( \limsup \) of the cocycle applied to \( u \) at \( x \) equals that at \( \varphi (t_0)x \) applied to the pushed-forward vector \( A(t_0,x)\, u \).
First, a.e. the discrete growth average \( n^{-1}\log \left\lVert \mathrm{cocycle}\, n\, x\, u \right\rVert \) has bounded range, with upper and lower bounds from the Furstenberg–Kesten Fekete inequalities \( \log \left\lVert \mathrm{cocycle} \right\rVert \le \mathrm{birkhoffSum}(\log ^{+}\left\lVert A(1,\cdot ) \right\rVert ) \) (and the inverse version), whose Birkhoff averages converge by the ergodic theorem. The cocycle identity and 8.4 give the shift relation \( \mathrm{cocycle}\, n\, (\varphi (t_0)x)\cdot A(t_0,x) = A(t_0,\varphi (n)x)\cdot \mathrm{cocycle}\, n\, x \), so the two growth averages differ by \( n^{-1}\bigl(\log \left\lVert A(t_0,\varphi (n)x)(\cdots ) \right\rVert - \log \left\lVert \cdots \right\rVert \bigr) \), which is squeezed to \( 0 \) by 8.10. A difference tending to \( 0 \) between two range-bounded sequences leaves the \( \limsup \) unchanged.
Let \( V \) be the Oseledets filtration of the time-one data, and fix \( t_0 \in \mathbb {R}\). Then for almost every \( x \) and every level \( i \),
Use the growth characterization \( v \in V_i\, x \iff v = 0 \lor \limsup \le \lambda _i \) at \( x \) and, pulled back along the measure-preserving \( \varphi (t_0) \), at \( \varphi (t_0)x \). The map \( P = A(t_0,x) \) is a bijection with inverse \( A(t_0,x)^{-1} \). For a non-bottom level, prove both inclusions by \( \mathrm{le\_ antisymm} \): membership of \( v \) (resp. \( P^{-1}v \)) is transported through the equivalence of growth \( \limsup \)s supplied by 8.11. The bottom level \( V_k = \bot \) maps to \( \bot \).
8.6 The continuous-flow theorem
Let \( \mu \) be a probability measure on \( X \), let \( \varphi \) be a measure-preserving one-parameter flow whose time-one map \( \varphi (1) \) is \( \mu \)-ergodic, and let \( A \) be a continuous-time linear cocycle over \( \varphi \) valued in invertible \( d \times d \) real matrices. Suppose \( g, g' \in L^1(\mu ) \) satisfy, for all \( s \in [0,1] \) and all \( x \),
Then there exist \( k \in \mathbb {N}\), a strictly decreasing sequence of exponents \( \lambda : \mathrm{Fin}\, k \to \mathbb {R}\), and a measurable family of subspaces \( V : \mathrm{Fin}(k+1) \to X \to \mathrm{Submodule}\, \mathbb {R}\, (\mathrm{EuclideanSpace}\, \mathbb {R}\, (\mathrm{Fin}\, d)) \) such that:
\( \lambda \) is strictly decreasing and each \( V_i \) is a measurable subspace family;
(full flow equivariance) for every \( t \in \mathbb {R}\), almost every \( x \) has \( A(t,x) \cdot V_i\, x = V_i\, (\varphi (t)x) \) for all \( i \);
almost every \( x \) carries the strict flag \( \top = V_0\, x \supsetneq \cdots \supsetneq V_k\, x = \bot \), and on each stratum \( v \in V_i\, x \setminus V_{i+1}\, x \) the continuous-time growth rate is exactly \( \lambda _i \):
\[ t^{-1}\log \left\lVert A(t,x)v \right\rVert \to \lambda _i \quad (t \to \infty ). \]
Take \( (k, \lambda , V) \) from the reduction 8.7; this supplies strict antitonicity, measurability, the strict flag, and the integer-time growth rates. Full flow equivariance at each \( t \) is 8.12. For the continuous-time growth, work a.e. on the discrete-conclusion set intersected with the error-sublinearity set of 8.8. Fix a stratum vector \( v \neq 0 \); rewriting via the reduction identity 8.4 turns the discrete stratum growth \( n^{-1}\log \left\lVert \mathrm{cocycle}\, n\, x\, v \right\rVert \to \lambda _i \) into \( n^{-1}\log \left\lVert A(n,x)v \right\rVert \to \lambda _i \). The between-times sandwich 8.9 then upgrades this integer-time limit to the continuous-parameter limit \( t^{-1}\log \left\lVert A(t,x)v \right\rVert \to \lambda _i \).