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.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 [1]) for the case of a flow (the "time" $t$ running through the real numbers). He proved (see [1], [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; regarding more general groups of transformations see, for example, [3] and [4]). Birkhoff called this property of $w$ (and its trajectory) recurrence; another terminology, suggested by W.H. Gottschalk and G.A. Hedlund [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 [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, [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 [2]–[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, [7]). Minimal sets are the fundamental objects of study in topological dynamics.

References

[1]	G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. (1927) MR1555257 Zbl 53.0733.03 Zbl 53.0732.01
[2]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) MR0121520 Zbl 0089.29502
[3]	W.H. Gottschalk, G.A. Hedlund, "Topological dynamics" , Amer. Math. Soc. (1955) MR0074810 Zbl 0067.15204
[4]	I.U. Bronshtein, "Extensions of minimal transformation groups" , Sijthoff & Noordhoff (1979) (Translated from Russian) MR0550605
[5]	B.M. Levitan, V.V. Zhikov, "Almost-periodic functions and differential equations" , Cambridge Univ. Press (1982) (Translated from Russian) MR0690064 Zbl 0499.43005
[6]	P. Hartman, "Ordinary differential equations" , Wiley (1964) MR0171038 Zbl 0125.32102
[7]	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 MR0584389 Zbl 0399.28011

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 [1], [2], [a8] on the one hand, and by [3], [a1], [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 [3], [4], [a1], [a2], and almost recurrent in [2] and [a8]. (In [2], [a8], almost periodicity has another meaning.) Formally, the notion of a recurrent point as defined in [1], [2], [a8] is different: see Recurrent point; a recurrent point is always [3]-almost periodic (i.e., almost-recurrent), but not conversely. In a dynamical system on a complete metric space the two notions coincide. (In [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 [1], [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 [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); see [4], [a2] and [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 [a4]. A Klein bottle cannot be minimal under a continuous flow (the Kneser theorem, see also [a6]), neither can the real projective plane (see [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 [a7] for references (the Seifert conjecture states that any smooth flow on $S^3$ has a periodic orbit; there is a $C^1$-counterexample, [a9]). For results about cascades, see [a3], [a10].

References

[a1]	J. Auslander, "Minimal flows and their extensions" , North-Holland (1988) MR0956049 Zbl 0654.54027
[a2]	R. Ellis, "Lectures on topological dynamics" , Benjamin (1969) MR0267561 Zbl 0193.51502
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[a4]	C. Gutierrez, "Smoothing continuous flows on two-manifolds and recurrences" Ergod. Th. Dynam. Sys. , 6 (1986) pp. 17–44 MR0837974 Zbl 0606.58042
[a5]	P.-F. Lam, "Inverses of recurrent and periodic points under homomorphisms of dynamical systems" Math. Systems Theory , 6 (1972) pp. 26–36 MR0301718 Zbl 0229.54035
[a6]	N.G. Markley, "The Poincaré–Bendixson theorem for the Klein bottle" Trans. Amer. Math. Soc. , 135 (1969) pp. 159–165 MR234442 Zbl 0175.50101
[a7]	L. Markus, "Lectures in differentiable dynamics" , Amer. Math. Soc. (1980) pp. Appendix II MR0309152 Zbl 0214.50701
[a8]	K.S. Sibirskii, "Introduction to topological dynamics" , Noordhoff (1975) (Translated from Russian) MR0357987
[a9]	P.A. Schweitzer, "Counterexamples to the Seifert conjecture and opening closed leaves of foliations" Amer. of Math. (2) , 100 (1974) pp. 386–400 MR0356086 Zbl 0295.57010
[a10]	A. Fahti, M. Herman, "Existence de diffeomorphismes minimaux" Astérisque , 49 (1977) pp. 37–59 MR482843