Let $V$ be a compact Riemannian manifold, $G$ the set of diffeomophisms of $V$, let $\nu$ be a probability measure in $G$. Suppose that $\exp_{x}$ is diffeomorphism in $\mathcal{B}_{2I}(x)\subset V$ ball of radius $2I$ for all $x\in V$, then we define $$ \begin{array}{rl} \delta_{1}(T) & ={\displaystyle \sup\left\{\left.\frac{d\left(T(x),T(y)\right)}{\left\|\exp_{x}^{-1}y\right\|} \: \right| \: (x,y)\in V^{2}, \:\: 0<d(x,y)\leq I \right\} } \\ \delta_{2}(T) &= {\displaystyle \sup\left\{ \left. \frac{\left\| \exp_{T(x)}^{-1}T(y)-DT(x)\exp_{x}^{-1}y \right\|}{\left\| \exp_{x}^{-1}y \right\|} \:\right|\: (x,y)\in V^{2},\: 0<d(x,y)\leq I,\: d(T(x),T(y))\leq I \right\} }. \end{array} $$ Now suppose that: $$ (H) \qquad \int \log^{+}\left( \delta_{1}(T)+\delta_{2}(T) \right)\nu (dT) <\infty. $$ Let $\Omega=G^{\mathbb{Z}}$, this is measurable space with product measure $\mathbb{P}=\nu^{\otimes \mathbb{Z}} $, and we consider the projection $T_{i}:\Omega\rightarrow G$, that is, if $\tau$ is the right translation then $T_{i}(\tau\omega)=T_{i+1}(\omega)$.

Show that $$\lim_{n\rightarrow \infty}\frac{1}{n}\log^{+}\left( \delta_{1}(T_{n})+\delta_{2}(T_{n}) \right)=0. \tag{1}$$

**Remark:** This question arises because I am trying to understand the article in this link (see pages 6,7), the author says that (1) is a consequence of the hypothesis (H) and the Birkhoff's ergodic theorem. I do not see how (1) you can follow these two facts, I would appreciate someone helping me to understand it.