Linear cocycle over a dynamical system

For a probability-preserving transformation $T$ , a measurable map $A:X\to GL_d(\mathbb R)$ generates the linear cocycle

A^{(0)}(x)=I,\qquad A^{(n)}(x)=A(T^{n-1}x)\cdots A(x)\quad(n\geq1).

It satisfies $A^{(n+m)}(x)=A^{(m)}(T^nx)A^{(n)}(x)$ . If $T$ is invertible, negative times are defined by $A^{(-n)}(x)=A^{(n)}(T^{-n}x)^{-1}$ .

Tangent dynamics

For a differentiable map, the chain rule makes its derivative iterates a cocycle on the tangent bundle. In a measurable trivialization this has the matrix form above. It describes growth of tangent vectors, whereas Koopman operators describe functions of the state.