  We employ techniques from optimal transport in order to prove decay of
transfer operators associated to iterated functions systems and expanding maps,
giving rise to a new proof without requiring a Doeblin-Fortet (or Lasota-Yorke)
inequality. Our main result is the following. Suppose $T$ is an expanding
transformation acting on a compact metric space $M$ and $A: M \to \mathbb{R}$ a
given fixed H{\"o}lder function, and denote by $L$ the Ruelle operator
associated to $A$. We show that if $L$ is normalized (i.e. if $L(1)=1$), then
the dual transfer operator $L^*$ is an exponential contraction on the set of
probability measures on $M$ with the $1$-Wasserstein metric.Our approach is
flexible and extends to a relatively general setting, which we name Iterated
Contraction Systems. We also derive from our main result several dynamical
consequences; for example we show that Gibbs measures depends in a
Lipschitz-continuous way on variations of the potential.
