Let $X$ be a compact metric space. Given a sequence $x_n \in X$ and an ultrafilter $\mathcal{U}$ on $\mathbb{N}$, we can define the "Banach limit" of ${x_n}$ with respect to $\mathcal{U}$. This limit is the unique element of $X$ such that every neighborhood of it contains a $\mathcal{U}$-large collection of the sequence ${x_n}$. However, this process is not necessarily translation-invariant. When $X$ is a closed interval in the reals, then one can define a Banach limit by using an ultrafilter and taking the ultrafilter-limit of the Cesaro means.
Is there a way to define a translation-invariant "limit" with reasonable properties for sequences taking values in an arbitrary compact metric space?