Kingman's subadditive ergodic theorem

Remarks

If $f_n = \sum_{k=0}^{n-1} f \circ T^k$ for an integrable $f$, then equality holds in the subadditivity condition, and the theorem reduces to Birkhoff's ergodic theorem. The integrals $a_n = \int_X f_n \, d\mu$ form a subadditive sequence, $a_{m+n} \le a_m + a_n$, so $a_n / n \to \gamma$ by Fekete's lemma; the theorem is the pointwise version of this fact. In the proof, the function $\liminf_n f_n / n$ is shown to be invariant. A covering argument cuts a long orbit segment into blocks on which $f_k / k$ is close to this $\liminf$. Subadditivity then bounds $f_N$ by the sum over the blocks.

Example: let $A \colon X \to GL_d(\R)$ be measurable with $\log^+ \norm{A}$ and $\log^+ \norm{A^{-1}}$ integrable, and let $f_n(x) = \log \norm{A(T^{n-1}(x)) \cdots A(T(x)) A(x)}$. Submultiplicativity of the operator norm gives $f_{m+n} \le f_m + f_n \circ T^m$, and the two integrability conditions make each $f_n$ integrable. For ergodic $T$, the theorem gives the almost sure limit of $f_n / n$, the top Lyapunov exponent of the matrix cocycle; this is the Furstenberg–Kesten theorem.

Going deeper

For a student

Kingman proved the theorem in 1968. It extends Birkhoff's theorem from additive to subadditive sequences. It is the main tool for the existence of Lyapunov exponents, as in the Furstenberg–Kesten theorem and in the proof of the Oseledets multiplicative ergodic theorem. Another standard use is the time constant of first-passage percolation, where travel times between points are subadditive but not additive.