Let $(X,d)$ be a metric space and $f:X\rightarrow X$ be continuous. Then, is there any meaning/research done on the metric
$$
D(x,y)\triangleq \sum\_{n \in \mathbb{N}} \frac1{2^n} d(f^n(x),f^n(y));
$$
where $f^0(x)=x$?



Orbit-based metric
In the linear case (for a bounded operator $T$ on a Banach space), the analogous construction for equivalent norms, possibly with a different ratio in place of $1/2$, is often named *adapted norm* (see e.g. *Global stability of dynamical systems*, by Michael Shub). Its main feature is that gives $T$ an operator norm arbitrarily close to its spectral radius. For metric spaces, note that $f$ is $2$-Lipschitz wrto $D$, and that the constant $2$ may be replaced to any $\theta>1$. At least, this shows that for a continuous self-map on a metric space, or on a complete metric space, being Lipschitz is not that special, if the constant is larger that $1$.


