The subadditive ergodic theorem

Let T preserve a probability measure and be ergodic. Let each real-valued function g-n be integrable, and assume the cocycle subadditivity inequality for every positive pair of lengths, outside a common null set. A convenient sufficient lower bound is g-n at least minus C times n for one finite deterministic constant C. Under these hypotheses, Kingman's theorem gives a finite constant limit of g-n divided by n almost everywhere. The limit equals the infimum of the normalized integrals. Invertibility and independence are not required.

The linear lower bound is stronger than necessary, but deliberately prevents the limit from being minus infinity and controls negative tails. Standard more general versions can allow a minus-infinite limit when lower control is absent. Under our stated assumptions convergence also holds in L1: subadditivity bounds the positive part of g-n divided by n by averages of shifts of the integrable positive part of g-one. These averages are uniformly integrable, while the negative parts are bounded by C. Together with almost-sure convergence this yields mean convergence. This additional argument explains the stronger conclusion rather than silently attaching it to every version.

Integrate the subadditive inequality and use preservation to remove the shift from the second integral. The resulting sequence of integrals is numerically subadditive. Fekete's lemma therefore identifies its normalized limit with its infimum. Kingman's deeper assertion identifies the same number as the almost-everywhere trajectory rate. The infimum is over integrals, not over the pointwise numbers at a chosen starting state. Moving an infimum through an integral or claiming the limit equals a pointwise infimum would generally change the statement.

The proof cannot simply divide a trajectory into blocks and assume those blocks are independent. Even when T is ergodic, a power of T used to sample nonoverlapping blocks need not be ergodic. Correct proofs handle shifted block offsets or invariant conditional limits carefully. Without ergodicity the limiting rate is an invariant function rather than necessarily a global constant. Thus subadditivity controls concatenation, while ergodicity controls whether different invariant regions can retain different asymptotic rates.

For matrix products the logarithm of an operator norm is subadditive because norms multiply as an upper bound. If matrices and their inverses have uniformly bounded norms, a deterministic linear lower bound follows, so the sufficient theorem above applies. More general integrable cocycles use the corresponding broader theorem. The growth exponent measures exponential scale only. A sequence of norms growing like a polynomial can have exponent zero while remaining unbounded, and strong one-step expansion can be canceled by later matrices.

The matrix instrument selects a single upper-triangular matrix and reports its determinant and singular stretches. Its animation interpolates geometry; it does not repeatedly multiply the selected matrix. To investigate a cocycle, compute its products separately and use the instrument only to check specified individual matrices. This distinction keeps finite-time singular values separate from a proven asymptotic rate. Long numerical records can suggest a limit, but without hypotheses and a theorem they do not establish almost-sure convergence or identify all exceptional trajectories.