View source for Minimal set

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A minimal set in a Riemannian space is a generalization of a [[Minimal surface|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|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|Plateau problem]] (cf. also [[Plateau problem, multi-dimensional|Plateau problem, multi-dimensional]]).

''A.T. Fomenko''

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 [[#References|[1]]]) for the case of a flow (the "time" $t$ running through the real numbers). He proved (see [[#References|[1]]], [[#References|[2]]]) 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|cascade]]; regarding more general groups of transformations see, for example, [[#References|[3]]] and [[#References|[4]]]). Birkhoff called this property of $w$ (and its trajectory) recurrence; another terminology, suggested by W.H. Gottschalk and G.A. Hedlund [[#References|[3]]], 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 [[#References|[5]]]). 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, [[#References|[6]]]). In the general case the structure of a minimal set can be very complicated (in this connection, in addition to what is said in [[#References|[2]]]–[[#References|[4]]] 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, [[#References|[7]]]). Minimal sets are the fundamental objects of study in [[Topological dynamics|topological dynamics]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. (1927) {{MR|1555257}} {{ZBL|53.0733.03}} {{ZBL|53.0732.01}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) {{MR|0121520}} {{ZBL|0089.29502}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W.H. Gottschalk, G.A. Hedlund, "Topological dynamics" , Amer. Math. Soc. (1955) {{MR|0074810}} {{ZBL|0067.15204}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.U. Bronshtein, "Extensions of minimal transformation groups" , Sijthoff &amp; Noordhoff (1979) (Translated from Russian) {{MR|0550605}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> B.M. Levitan, V.V. Zhikov, "Almost-periodic functions and differential equations" , Cambridge Univ. Press (1982) (Translated from Russian) {{MR|0690064}} {{ZBL|0499.43005}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> P. Hartman, "Ordinary differential equations" , Wiley (1964) {{MR|0171038}} {{ZBL|0125.32102}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> A.B. Katok, Ya.G. Sinai, A.M. Stepin, "Theory of dynamical systems and general transformation groups with invariant measure" ''J. Soviet Math.'' , '''7''' (1977) pp. 974–1065 ''Itogi. Nauk. i Tekhn. Mat. Anal.'' , '''13''' (1975) pp. 129–262 {{MR|0584389}} {{ZBL|0399.28011}} </TD></TR></table>

''D.V. Anosov''

====Comments====
The terminology around the notions of recurrence and almost periodicity of points in a topological dynamical system is confusing. These are two mainstreams of nomenclature, represented by [[#References|[1]]], [[#References|[2]]], [[#References|[a8]]] on the one hand, and by [[#References|[3]]], [[#References|[a1]]], [[#References|[a2]]] on the other. The type of point mentioned above, namely, a point $w$ such that for every neighbourhood $U$ of $w$ the set $\{t\colon S_tw\in U\}$ is relatively dense in $\mathbf R$, is called almost periodic in [[#References|[3]]], [[#References|[4]]], [[#References|[a1]]], [[#References|[a2]]], and almost recurrent in [[#References|[2]]] and [[#References|[a8]]]. (In [[#References|[2]]], [[#References|[a8]]], almost periodicity has another meaning.) Formally, the notion of a recurrent point as defined in [[#References|[1]]], [[#References|[2]]], [[#References|[a8]]] is different: see [[Recurrent point|Recurrent point]]; a recurrent point is always [[#References|[3]]]-almost periodic (i.e., almost-recurrent), but not conversely. In a dynamical system on a complete metric space the two notions coincide. (In [[#References|[3]]] the notion of a recurrent point is used in the meaning of "positively and negatively Poisson stable".) What Birkhoff proved was the equivalence of recurrence of a point $w$ (according to the terminology of [[#References|[1]]], [[#References|[2]]]) and the property that $w$ has a compact minimal orbit closure, provided the phase space is a complete metric space. Using the terminology of [[#References|[3]]] one can show: If $w$ has a compact minimal orbit closure, the $w$ is an almost-periodic point (no conditions on the phase space); conversely, an almost-periodic point has a minimal orbit closure, which is compact if the phase space is locally compact and Hausdorff (no metrizability assumed).

The classification of compact minimal sets in topological dynamics is a largely unsolved problem. Only for special classes something can be said (cf. [[Distal dynamical system|Distal dynamical system]]); see [[#References|[4]]], [[#References|[a2]]] and [[#References|[a1]]]. Unsolved is also the problem as to which (compact) Hausdorff spaces can be the phase space of a minimal flow or a minimal cascade. In this respect, Schwarz's theorem, mentioned above, gives a partial solution for compact surfaces; for a generalization, see [[#References|[a4]]]. A Klein bottle cannot be minimal under a continuous flow (the [[Kneser theorem|Kneser theorem]], see also [[#References|[a6]]]), neither can the real projective plane (see [[#References|[a5]]]). Still open is Gottschalk's conjecture (a particular case of Seifert's conjecture): $S^3$ cannot be the phase space of a minimal flow; see Appendix II of [[#References|[a7]]] for references (the Seifert conjecture states that any smooth flow on $S^3$ has a periodic orbit; there is a $C^1$-counterexample, [[#References|[a9]]]). For results about cascades, see [[#References|[a3]]], [[#References|[a10]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Auslander, "Minimal flows and their extensions" , North-Holland (1988) {{MR|0956049}} {{ZBL|0654.54027}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Ellis, "Lectures on topological dynamics" , Benjamin (1969) {{MR|0267561}} {{ZBL|0193.51502}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> G. Glasner, B. Weiss, "On the construction of minimal skew products" ''Israel J. Math.'' , '''34''' (1979) pp. 321–336 {{MR|0570889}} {{ZBL|0434.54032}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> C. Gutierrez, "Smoothing continuous flows on two-manifolds and recurrences" ''Ergod. Th. Dynam. Sys.'' , '''6''' (1986) pp. 17–44 {{MR|0837974}} {{ZBL|0606.58042}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> P.-F. Lam, "Inverses of recurrent and periodic points under homomorphisms of dynamical systems" ''Math. Systems Theory'' , '''6''' (1972) pp. 26–36 {{MR|0301718}} {{ZBL|0229.54035}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> N.G. Markley, "The Poincaré–Bendixson theorem for the Klein bottle" ''Trans. Amer. Math. Soc.'' , '''135''' (1969) pp. 159–165 {{MR|234442}} {{ZBL|0175.50101}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> L. Markus, "Lectures in differentiable dynamics" , Amer. Math. Soc. (1980) pp. Appendix II {{MR|0309152}} {{ZBL|0214.50701}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> K.S. Sibirskii, "Introduction to topological dynamics" , Noordhoff (1975) (Translated from Russian) {{MR|0357987}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> P.A. Schweitzer, "Counterexamples to the Seifert conjecture and opening closed leaves of foliations" ''Amer. of Math. (2)'' , '''100''' (1974) pp. 386–400 {{MR|0356086}} {{ZBL|0295.57010}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> A. Fahti, M. Herman, "Existence de diffeomorphismes minimaux" ''Astérisque'' , '''49''' (1977) pp. 37–59 {{MR|482843}} {{ZBL|}} </TD></TR></table>