Multiplicative ergodic theorem and Lyapunov spectra

Let T be an invertible bimeasurable ergodic probability-preserving transformation on a standard Borel probability space, and let A be a measurable map into invertible real d-by-d matrices. Assume the positive parts of the natural logarithms of the operator norms of A and its inverse are integrable. These are sufficient hypotheses for the two-sided finite-dimensional theorem used here. The cocycle product applies the earliest matrix first, on the right. Reversing the order can change a product when matrices do not commute.

The theorem gives finitely many finite real exponents with multiplicities and a measurable direct-sum decomposition of the vector space at almost every base point. Applying A sends each subspace to its corresponding subspace over the next point. Every nonzero vector in one subspace has its assigned exponential growth rate. In the ergodic case the exponents and dimensions are almost surely constant; without ergodicity they can be invariant measurable functions. Almost-sure statements do not automatically cover all initial base points or supply numerical convergence rates.

A vector with components in several Oseledets subspaces has the largest exponent among its nonzero components. Singular values of the full n-step product also have normalized logarithms converging to the exponents, repeated with multiplicity. However, singular vectors of a particular finite product are optimized for that finite interval and need not equal the equivariant splitting directions. This distinction is already visible for a constant nonnormal matrix, where a coordinate direction can feed an expanding eigenvector despite having a contracting diagonal entry.

With only forward integrability, a more general theorem provides a forward filtration, and some exponents may equal minus infinity. One should not claim the full invertible splitting stated above after dropping inverse integrability or invertibility of the base. Different formulations address noninvertible bases, but require their own conclusions and hypotheses. Here the explicit assumptions keep forward and backward growth compatible and make every exponent finite. A small observed singular value at one time is not by itself evidence of a negative limiting exponent.

The largest stretch of an exterior power measures growth of oriented volume in the corresponding dimension. Applying the subadditive theorem to these exterior-power cocycles identifies sums of the largest exponents; differences of consecutive sums recover individual values. The top exterior power is determinant, whose logarithm is additive along products. Birkhoff then gives the displayed sum rule. This explains why determinant one forces the sum of exponents to vanish but does not force each exponent to vanish. Expansion and contraction can compensate.

The matrix lab's family has bottom-right entry one, so it does not directly represent every matrix in the worked example. It can nevertheless expose the difference between one-step stretching and asymptotic growth for its own selected constant matrix. Its playback interpolates geometry rather than composing a cocycle. Compute powers and their normalized logarithms separately, and treat the readouts as checks on an individual matrix. The theorem establishes the long-time conclusion only after its measure, measurability, and integrability assumptions have been verified.