# Minimal set

A minimal set in a Riemannian space is a generalization of a minimal surface. A minimal set is a $k$-dimensional closed subset $X_0$ in a Riemannian space $M^n$, $n>k$, such that for some subset $Z$ of $k$-dimensional Hausdorff measure zero the set $X_0\setminus Z$ is a differentiable $k$-dimensional minimal surface (that is, is an extremum of the $k$-dimensional volume functional $\Lambda^k$, defined on $k$-dimensional surfaces imbedded in $M^n$). The notion of a "minimal set" amalgamates several mathematical ideas called upon to serve in the so-called multi-dimensional Plateau problem (cf. also Plateau problem, multi-dimensional).
A minimal set in a topological dynamical system $\{S_t\}$ is a non-empty closed invariant (that is, consisting wholly of trajectories) subset $F$ of the phase space $W$ of the system which does not have proper closed invariant subsets. The latter is equivalent to saying that each trajectory in $F$ is everywhere dense in $F$. The notion of a minimal set was introduced by G.D. Birkhoff (see ) for the case of a flow (the "time" $t$ running through the real numbers). He proved (see , ) that if $F$ is a compact minimal set and $w\in F$, then for any neighbourhood $U$ of $w$ the set of those $t$ for which $S_tw\in U$ is relatively dense in $\mathbf R$ (that is, there is an $l$ such that in each "time interval" $[s,s+l]$ of length $l$ there is at least one $t$ with $S_tw\in U$); conversely, if $W$ is a complete metric space and a point $w$ has the above property, then the closure of its trajectory $\{S_tw\}$ is a compact minimal set (the same is true for a cascade; regarding more general groups of transformations see, for example,  and ). Birkhoff called this property of $w$ (and its trajectory) recurrence; another terminology, suggested by W.H. Gottschalk and G.A. Hedlund , is also used, in which this property is called almost-periodicity of the point $w$. If $F=W$, then the dynamical system itself is called minimal.
If a trajectory has compact closure, then it contains a minimal set $F$ (for semi-groups of continuous transformations $\{S_t\}$, with non-negative real or integer $t$, an analogue of this result holds, where in $F$ the transformations $S_t$ are even invertible ). However, research into the limit behaviour of the trajectories of a dynamical system does not reduce to the study of only the minimal sets of the latter. A minimal set of a smooth flow of class $C^2$ on a two-dimensional closed surface $S$ has a very simple structure: it is either a point, a closed trajectory or the whole surface, which is then a torus (Schwarz's theorem, ). In the general case the structure of a minimal set can be very complicated (in this connection, in addition to what is said in  it must be said that minimality of a dynamical system places no restrictions on its ergodic properties with respect to any of its invariant measures, ). Minimal sets are the fundamental objects of study in topological dynamics.