Let $(X,\mathcal B,\mu)$ be a probability space, let $T:X\to X$ be an invertible ergodic measure-preserving map, and let $A:X\to GL(d,\mathbb R)$ be measurable with
so the negative iterates use the inverse cocycle along the backward orbit of the invertible base map, and the dimensions $m_i=\dim E_i(x)$ are constant for $\mu$-a.e. $x$.