affine IFS, we don’t assume that $M_j$ are invertible or contracting (in the sense that $\|M_j\|<1$ where $\|\cdot\|$ is the matrix operator norm). We say that ${\mathcal S}$ is [*contracting*]{} if all $M_j$ are contracting. It is well-known that if $\mathcal S$ is contracting, there exists a unique non-empty compact set $K\subset \R^d$ such that $$K=\bigcup_{j\in \Lambda} S_j(K).$$ We call $K$ the [*self-affine set*]{} generated by ${\mathcal S}$. In particular, if all the maps in ${\mathcal S}$ are contracting similitudes, we call $K$ a [*self-similar set*]{}. Let $(\Sigma,\sigma)$ be the two-sided full shift over the alphabet $\Lambda$, i.e. $\Sigma=\Lambda^\Z$ and $\sigma:\Sigma\to \Sigma$ is the left shift map. Endow $\Sigma$ with the discrete product topology and let $\M_\sigma(\Sigma)$ denote the space of $\sigma$-invariant Borel probability measures on $\Sigma$.

\[de-1.1\] Let $m\in \M_\sigma(\Sigma)$. An affine IFS ${\mathcal S}=\{M_jx+a_j\}_{j\in \Lambda}$ is said to be average contracting with respect to $m$ if, for $m$-a.e. $x=(x_n)_{n=-\infty}^\infty\in\Sigma$, the top Lyapunov exponent $\lambda(x)$ defined by $$\label{e-lyapunov}
\lambda(x)=\lim_{n\to \infty} \frac{1}{n}\log \|M_{x_{0}}\cdots M_{x_{n-1}}\|$$ is strictly negative.

We remark that the above limit in defining $\lambda(x)$ exists and takes values in $[-\infty, \infty)$ for $m$-a.e. $x$. This follows from the Furstenberg-Kesten theorem [@FurstenbergKesten1960] or Kingman’s sub-additive ergodic theorem [@Kingman1968].

Now 