A stochastic matrix $P$ is a $n \times n-$matrix with entries $p_{ij} \in [0,1]$ so that $\sum_{k=1}^n p_{ik} = 1$ for every $i \in \{1,...,n\}$. The matrix $P$ is called positive, if no entry $p_{ij}$ is zero. Now we want to find a metric $d$ on the probability simplex $$ D := \{x \in \mathbb{R}^n \vert \ x_i \ge0 \ \ \land \ \sum_{k=1}^n x_k = 1 \}$$ so that, for every positive stochastic matrix $P$, the map $$ \varphi_P : D \rightarrow D, \quad \quad \varphi_D(x)=P^{T}x $$ is a contraction.
Since for every stochastic matrix $P$ the middle point $m=(1,...,1)\in D$ of the simplex is a fixed point, I thought it would be a good start to construct the metric using $m$ as a kind of a "center". I defined a metric $d$ via
$$d : D \times D \rightarrow \mathbb{R}$$ $$d(x,y) = \begin{cases} \vert \vert x-y \vert \vert /n \quad \text{if x,y and m are in a line} \\ (\vert \vert x-m \vert \vert +\vert \vert y-m \vert \vert)/n \quad \text{else} \end{cases}$$
Here $\vert \vert x-y \vert \vert$ is the euclidean distance. Basically, $d$ measures the distance from $x$ to $y$ if you cannot go directly from one point to another but if you always have to cross the point $m$. I think I showed that this is indeed a metric, and I also showed that $\varphi_P$ is a contraction in the case if $x,y$ and $m$ are in a line. But so far I failed to show the contraction property for the other case. Have I not found the right metric yet? I'm thankful for any advise.