  This paper studies the Yang--Mills ASD equation over the cylinder as a
non-linear evolution equation. We consider a dynamical system consisting of
bounded orbits of this evolution equation. This system contains many chaotic
orbits, and moreover it becomes an infinite dimensional and infinite entropy
system. We study the mean dimension of this huge dynamical system. Mean
dimension is a topological invariant of dynamical systems introduced by Gromov.
We prove the exact formula of the mean dimension by developing a new technique
based on the metric mean dimension theory of Lindenstrauss--Weiss.
