Subadditive cocycle - Knowlpedia

A subadditive cocycle over $T$ is a sequence of measurable real functions $a_n$ , $n\geq0$ , with $a_0=0$ and

a_{n+m}(x)\leq a_n(x)+a_m(T^nx) \quad(n,m\geq0)

almost everywhere. Equality defines an additive cocycle.

Examples

Orbit sums $a_n=\sum_{j=0}^{n-1}f\circ T^j$ are additive. For products $A^{(n)}(x)=A(T^{n-1}x)\cdots A(x)$ of invertible matrices, $a_n(x)=\log\|A^{(n)}(x)\|$ is subadditive by the operator-norm product inequality.