transformations and derive a useful result (Lemma \[lem-inderg\]). In the end, we present a version of Oseledets’ multiplicative ergodic theorem due to Froyland et al. [@FroylandLloydQuas2010].

Conditional information and entropy {#Sect-ent}
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Let $(X,\B,m)$ be a probability space. For a sub-$\sigma$-algebra $\A$ of $\B$ and $f\in L^1(X,\B,m)$, we denote by ${\bf E}_m(f|\A)$ the the [*conditional expectation of $f$ given $\A$*]{}. For a countable $\B$-measurable partition $\xi$ of $X$, we denote by ${\bf  I}_m(\xi|\A)$ the [*conditional information of $\xi$ given $\A$*]{}, which is given by the formula $$\label{e-1.2}
 {\bf I}_m(\xi|\A)=-\sum_{A\in \xi}\chi_A\log \E_m(\chi_A |\A),$$ where $\chi_A$ is the characteristic function on $A$. The [*conditional entropy of $\xi$ given $\A$*]{}, written as $H_m(\xi|\A)$, is defined by the formula $$H_m(\xi|\A)=\int {\bf I}_m(\xi|\A)\; dm.$$ (See e.g. [@Parry1981; @Walters1982] for more details.) The above information and entropy are unconditional when $\A={{\mathcal N}}$, the trivial $\sigma$-algebra consisting of sets of measure zero and one, and in this case we write $${\bf I}_m(\xi|{{\mathcal
N}})=:{\bf I}_\nu(\xi)\quad\mbox{and}\quad H_m(\xi|{{\mathcal
N}})=:H_m(\xi).$$

For a countable partition $\xi$, we use $\widehat{\xi}$ to denote the $\sigma$-algebra generated by $\xi$. If $\xi_1$,…, $\xi_n$ are countable partitions, then $\xi_1\vee \cdots \vee \xi_n=\bigvee_{i=1}^n \xi_i$ denotes the partition consists of the sets $A_1\cap\cdots \cap A_n$ with $A_i\in \xi_i$. Similarly 