Minimal dynamical systems

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Sergiy Kolyada and Ľubomír Snoha (2009), Scholarpedia, 4(11):5803. doi:10.4249/scholarpedia.5803 revision #128110 [link to/cite this article]
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Curator: Sergiy Kolyada

Minimal systems are natural generalizations of periodic orbits, and they are analogues of ergodic measures in topological dynamics. They were defined by G. D. Birkhoff in 1912 [Bir] as the systems which have no nontrivial closed subsystems ("nontrivial" means "non-empty and proper" where the word "proper" is used throughout the article in the meaning "not equal to the whole space"). Minimal systems can be considered to be the most fundamental dynamical systems. General references are [GH], [Got1], [Got2], [Ell1], [Br], [Au] and [Vri].


Minimal systems - equivalent definitions

By a dynamical system \((X,f)\) we mean a topological space \(X\) together with a continuous map \(f: X\to X\ .\) The space \(X\) is sometimes called the phase space of the system. A set \(A\subseteq X\) is called \(f\)-invariant if \(f(A)\subseteq A\ .\)

A dynamical system \((X,f)\) is called minimal if \(X\) does not contain any non-empty, proper, closed \(f\)-invariant subset. In such a case we also say that the map \(f\) itself is minimal. Thus, one cannot simplify the study of the dynamics of a minimal system by finding its nontrivial closed subsystems and studying first the dynamics restricted to them.

Given a point \(x\) in a system \((X,f)\ ,\) \(Orb_f(x)= \{x, f(x), f^2(x), ... \}\) denotes its orbit (by an orbit we mean a forward orbit even if \(f\) is a homeomorphism) and \(\omega_{f} (x)\) denotes its \(\omega\)-limit set, i.e. the set of limit points of the sequence \(x, f(x), f^2(x), ... \ .\) The following conditions are equivalent:

  • (\(X,f)\) is minimal,
  • every orbit is dense in \(X\ ,\)
  • \(\omega_{f} (x) = X\) for every \(x\in X\ .\)

A minimal map \(f\) is necessarily surjective if \(X\) is assumed to be Hausdorff and compact.

Examples of minimal homeomorphisms

Example 1. Consider a homeomorphism of the \(2\)-torus, \(S: \Bbb T \to \Bbb T\ ,\) of the form \(S(x,y)=(x+\alpha, y+\beta)\ ,\) where \(1,\alpha, \beta \in \Bbb R\) are rationally independent and \(+ : \Bbb R / \Bbb Z \times \Bbb R \to \Bbb R / \Bbb Z\) is defined in the obvious way. Then \(S\) is minimal (and ergodic with respect to Lebesgue measure). M. Rees [R1] found a minimal homeomorphism \(S_1\) which is an extension of \(S\) (i.e., \(\varphi \circ S_1 = S\circ \varphi\) for some continuous surjection \(\varphi\) of \(\Bbb T\)) such that \(S_1\) has positive topological entropy. In fact every \(n\)-manifold, \(n\geq 2\ ,\) which carries a minimal homeomorphism also carries a minimal homeomorphism with positive topological entropy [BCLR].

Example 2. Let \(K=(k_n)_{n>0}\) be a sequence of integers \(k_n\geq 2\ .\) Let \(\Sigma_{K}\) be the set of all one-sided infinite sequences \((i_n)_{n\geq 1}\) for which \(0\leq i_n \leq k_n-1\ .\) Think of these sequences as 'integers' in multibase notation, the base of the \(n^{th}\) digit \(i_n\) being \(k_n\ .\) With the natural (product) topology, \(\Sigma_{K}\) is homeomorphic to the Cantor set. Define a map \( \alpha_{K}: \Sigma_{K} \to \Sigma_{K}\) which informally may be described as 'add 1 and carry' where the addition is performed at the leftmost term \(i_1\) and the carry proceeds to the right in multibase notation. Then \(\alpha_{K}\) is a minimal homeomorphism and is called a generalized adding machine or an odometer (as a general reference see e.g. [Dow]).

In general it is difficult to construct a minimal homeomorphism, see for instance the examples of minimal homeomorphisms on the Klein bottle in [Ell] and [Par]. For some methods of constructions of minimal homeomorphisms see [AnK], [Ell], [GW] and [FK].

Existence of minimal sets

Given a dynamical system \((X,f)\ ,\) a set \(M\subseteq X\) is called a minimal set if it is non-empty, closed and invariant and if no proper subset of \(M\) has these three properties. So, \(M\subseteq X\) is a minimal set if and only if \((M, f|_M)\) is a minimal system. A system \((X,f)\) is minimal if and only if \(X\) is a minimal set in \((X,f)\ .\)

The basic fact discovered by G. D. Birkhoff is that in any compact system \((X,f)\) there are minimal sets. This follows immediately from the Zorn's lemma.

Since any orbit closure is invariant, we get that any compact orbit closure contains a minimal set. This is how compact minimal sets may appear in non-compact spaces. Two minimal sets in \((X,f)\) either are disjoint or coincide. A minimal set \(M\) is strongly \(f\)-invariant, i.e. \(f(M)=M\ ,\) provided it is compact Hausdorff.

Minimality and syndetical recurrence

A set \(A\subseteq \mathbb N\) is called syndetic if it has bounded gaps, i.e. if there exists \(N\in \mathbb N\) such that every block of \(N\) consecutive positive integers intersects \(A\ .\)

Given a dynamical system \((X,f)\ ,\) a point \(x\in X\) is said to be syndetically recurrent (or strongly recurrent or uniformly recurrent or almost periodic) if for every open neighborhood \(U\) of \(x\) the set of return times \(n(x,U) = \{n \geq 0:\, f^n(x)\in U\}\) is syndetic. Thus a syndetically recurrent point is one which is recurrent with `bounded return times'.

There is a closed connection between syndetical recurrence and minimal systems. Let \((X,f)\) be a dynamical system.

  • If \(X\) is compact and \((X,f)\) is minimal then every point \(x\in X\) is syndetically recurrent.
  • Conversely, if \(X\) is regular and \(x \in X\) is syndetically recurrent then its orbit closure \(\overline{Orb_f(x)}\) is a minimal set.

So, if the phase space \(X\) is regular and every point \(x \in X\) is syndetically recurrent then the system is the disjoint union of its minimal subsystems. Such systems are sometimes called semi-simple. A nice example of a semi-simple system \((X,f)\) is the unit disk rotated at different rates around the center. Precisely, in polar coordinates, let \(X\) be given by \(0 \leq r \leq 1\) and \(f(r, \vartheta) = (r, \vartheta +r)\ .\)

Minimality of a map and its iterates

A system \((X,f)\) is called totally minimal if \((X,f^n)\) is minimal for all \(n=1,2,\dots\ .\) We describe what happens if a system is minimal but not totally minimal.

Let \(X\) be a compact Hausdorff space and \(f: X\to X\) be continuous. If \(f\) is minimal but \(f^n\) is not, then there are pairwise disjoint compact subsets \(X_i \subseteq X\ ,\) uniquely defined up to the order, with \(X=X_0\cup X_1 \cup \dots \cup X_{k-1}\ ,\) such that \(k\geq 2\) is a divisor of \(n\ ,\) \(f(X_i)=X_{i+1(\mod k)}\) and \(f^n|_{X_i}\) is minimal for each \(i\) (hence also \(f^k|_{X_i}\) is minimal for each \(i\)). Since the minimal sets for \(f^n\) are uniquely defined and pairwise disjoint, they are just the sets \(X_0, X_1,\dots, X_{k-1}\ .\) In other words, the number of all distinct subsets of \(X\) minimal for \(f^n\) is equal to \(k\ .\)

As a corollary we get that if a compact Hausdorff space \(X\) is connected and \(f: X\to X\) is minimal then \(f\) is totally minimal.

For more details on this topic see [Ye], cf. [Ban].

Other equivalent definitions of a minimal system

For a compact metric space \(X\) and a continuous map \(f: X\to X\) the following are equivalent:

  • \((X,f)\) is minimal.
  • \(f(X)=X\) and every backward orbit of every point in \(X\) is dense (by a backward orbit of \(x_0\in X\) we mean any set \(\{x_0, x_1, ..., x_n, ...\}\) with \(f(x_{i+1})=x_i\) for \(i\geq 0\)).
  • The only closed subsets \(E\) of \(X\) with \(f(E)\supseteq E\) are \(\emptyset\) and \(X\ .\)
  • For every non-empty open set \(U\subseteq X\ ,\) there exists \(N\in \mathbb N\) such that \(\bigcup _{n=0}^N f^{-n}(U) = X\ .\)

Topological properties of minimal maps

A continuous map \(f:X\to Y\) between topological spaces is called irreducible if it is surjective and \(f(A)\ne Y\) for every proper closed subset \(A\subset X.\) A map \(f: X\to Y\) is called almost open if it sends non-empty open sets to sets with non-empty interior (the terminology is not unified -- instead of almost open some authors say semi-open, feebly open, somewhat open or quasi-interior).

Let \(X\) be a compact Hausdorff space and \(f: X \to X\) continuous. Then

  • \(f\) is minimal \(\Longrightarrow\) \(f\) is irreducible \(\Longrightarrow\) \(f\) is almost open

and if \(f\) is minimal then the following are equivalent:

  • \(f\) is open \(\Longleftrightarrow\) \(f\) is injective \(\Longleftrightarrow\) \(f\) is a homeomorphism.

It follows that any minimal map in a compact Hausdorff space is either a homeomorphism or a non-invertible and non-open (but irreducible and hence almost open) map.

Another interesting property of minimal maps in compact Hausdorff spaces is the following one:

  • For every non-empty open set \(U\subseteq X\ ,\) there exists \(N\in \mathbb N\) such that \(\bigcup _{n=0}^N f^{n}(U) = X\ .\)

Though minimal maps need not be invertible, in some aspects they behave like homeomorphisms. For instance, if \(f\) is a minimal map in a compact Hausdorff space \(X\) and \(A\subseteq X\) then both \(f(A)\) and \(f^{-1}(A)\) share some topological properties with the set \(A\) -- namely the ones which describe how large a set is. In fact, the following claims hold.

  • If \(A\) is nowhere dense (dense, of 1st category, of 2nd category, residual) then both \(f(A)\) and \(f^{-1}(A)\) are nowhere dense (dense, of 1st category, of 2nd category, residual), respectively.
  • If \(A\) has nonempty interior (has the Baire property) then both \(f(A)\) and \(f^{-1}(A)\) have nonempty interior (have the Baire property), respectively.
  • If \(A\) is open then there is an open set \(B\subseteq X\) such that \(B\subseteq f(A) \subseteq \overline{B}\) (here \(B\) may not be unique; the largest of such sets is always the interior of \(f(A)\)).

The fact that in some aspects minimal maps behave like homeomorphisms will be less surprising in the light of the following result.

Let \(X\) be a compact metric space and \(f: X\to X\) be minimal. Then

  • \(f\) is almost one-to-one, which means that the set \( \{x\in X : \# f^{-1}(x)=1\}\) is a \(G_{\delta}\)-dense set in \(X\ .\)
  • there exists a residual set \(Y\subseteq X\) such that \(f(Y)=Y\) and \(f|_Y\) is a minimal homeomorphism. Moreover, \((f|_Y)^{-1}\) is also a minimal homeomorphism and while \(f|_Y\) is uniformly continuous, \((f|_Y)^{-1}\) is uniformly continuous only in the case when \(f\) is a homeomorphism (then one can take \(Y=X\)).

For proofs of these results see [KST].

Examples of minimal non-invertible maps

To construct a minimal non-invertible map in a given space is usually more difficult than to construct a minimal homeomorphism. However, symbolic dynamics provides many examples of minimal non-invertible maps. Given a finite alphabet \(A\ ,\) consider \(A^{\Bbb N}\) together with the shift \(\sigma\ .\) One can prove that any subshift of \((A^{\mathbb N}, \sigma)\) on which the shift acts injectively consists of finitely many periodic orbits. Hence any minimal subshift of \((A^{\mathbb N}, \sigma)\) which is not reduced to a periodic orbit is non-invertible. On the other hand by the Jewett-Krieger theorem there exist a variety of minimal subshifts, most of which do not consist of a periodic orbit; among them, zero-entropy as well as positive-entropy systems with various properties. Other examples are less abstract: one-sided Sturmian and Toeplitz systems are minimal subshifts, none of which is reduced to one periodic orbit.

Other interesting examples of non-invertible minimal maps on a Cantor set come from interval dynamics when a suitable interval map is restricted to an invariant Cantor set. For instance, there are unimodal maps whose restriction to a Cantor set (the \(\omega\)-limit set \(\omega(c)\) of the critical point \(c\)) is minimal and fails to be invertible only at \(k\) points, each of them lying in the backward orbit of \(c\) (one of them is \(c\) itself) and having two preimages in \(\omega(c)\) (all other points in \(\omega(c)\) have only one preimage in \(\omega(c)\)), see [BKP].

The first examples of non-invertible minimal maps on a manifold, namely on the \(2\)-torus, were found in [KST]. Such a map can for instance be constructed by developing ideas from [R]. More generally, any minimal skew product homeomorphism of the torus having an asymptotic pair of points has an almost one-to-one factor which is a minimal non-invertible map of the torus.

Examples of non-invertible minimal maps in some more exotic spaces can be found in [BKS] and in [SS] (they are mentioned in the next section) and in [AY].

Spaces admitting minimal maps

The classification, i.e. the full topological characterization of compact metric spaces admitting minimal maps is a well-known open problem in topological dynamics, solved only in few particular cases.

If a space allows a minimal map, the proof usually builds on a standard example of a minimal homeomorphism (see Section 2). Proofs that a space does not admit any minimal map/homeomorphism often rely on the fixed (periodic) point property. For example, any homeomorphism on a compact manifold with non-zero Euler characteristic (homotopic to the identity or not) has a periodic point, hence all compact surfaces except the torus and the Klein bottle do not admit minimal homeomorphisms. One result which can be used if the space does not have the fixed point property is that if \(X\) is a non-compact Hausdorff topological space with a compact subset having non-empty interior, then \(X\) does not admit any minimal map (see [Got]).

There are spaces, even metric continua, of all four possible types from the point of view whether they admit a minimal homeomorphism or not and whether they admit a minimal non-invertible map or not. The \(2\)-torus admits both of them, the unit compact interval admits neither of them. The circle admits no minimal non-invertible map, while admitting a minimal homeomorphism. The pinched \(2\)-torus (i.e. the torus on which two points are identified) admits a minimal non-invertible map but it has a fixed point property for homeomorphisms. For other interesting examples of continua in this context see [BKS].

A necessary condition for a compact metric space X to admit a minimal map is that the quotient space \(X/C\ ,\) where \(C\) is the decomposition of \(X\) into the connected components, be either finite or Cantor. However, this condition is far from being sufficient.

The problem of the classification of spaces admitting minimal maps is solved in two important classes of spaces -- in the class of 2-manifolds and in the class of almost totally disconnected compact metric spaces.

First let us discuss manifolds. Suppose that \(f : \mathcal M^2 \to \mathcal M^2\) is a minimal map of a \(2\)-manifold (compact or not, with or without boundary). Then \(f\) is a monotone map with tree-like point inverses and \(\mathcal M^2\) is either a finite union of tori or a finite union of Klein bottles which are cyclically permuted by \(f\ ,\) see [BOT].

It is known (Church [Ch]) that any \(C^\omega\) (real analytic) monotone onto map on a compact connected \(n\)-manifold without boundary is a homeomorphism. Therefore there are no minimal non-invertible \(C^\omega\) maps on surfaces. The examples of minimal non-invertible maps on the \(2\)-torus (which are constructed in [KST]) are just \(C^0\) maps. So, the existence of smooth minimal non-invertible maps on manifolds is still an open problem.

On manifolds of dimension \(\geq 3\) a general theorem by Katok [Ka], and Fathi and Herman (see [FH]) ties the existence of minimal diffeomorphisms to the existence of locally free diffeomorphisms. In particular all the odd-dimensional spheres admit minimal diffeomorphisms. The classification of compact \(n\)-manifolds, \(n \geq 3\ ,\) admitting minimal maps is an open problem.

We are able to characterize spaces admitting minimal maps also among all almost totally disconnected compact metric spaces. A space \(X\) is said to be almost totally disconnected if the set of its degenerate components, considered as a subset of \(X\ ,\) is dense in \(X\) (a connected component is called degenerate if it is just one point). A compact metric space \(X\) is said to be a cantoroid if it is almost totally disconnected and has no isolated point. An almost totally disconnected compact metric space admits a minimal map if and only if it is either a finite set or a cantoroid (see [BDHSS]). This result shows, among others, that there exist many minimal systems with nonhomogeneous phase spaces (a space \(X\) is homogeneous, i.e. for any points \(x,y\in X\) there is a homeomorphism \(h: X\to X\) with \(h(x)=y\)). Examples of nonhomogeneous minimal systems on cantoroids are Floyd-Auslander systems (see, e.g., [HJ] and references therein), some non-invertible minimal systems which are generalizations of Floyd-Auslander systems (see [SS]) and some others.

Topological structure of minimal sets

The problem of understanding the behavior of all points of a given system under forward iteration and, in particular, finding all minimal sets of the system is central in topological dynamics. It seems that Dowker [Dowk] and Cartwright [Car] were the first who studied the topological structure of minimal sets (of homeomorphisms). Since then it has been a topic of constant interest.

Much is known on the topological structure of minimal sets in spaces with dimension at most one. If \(X\) is a compact zero-dimensional space, \(f : X \to X\) is continuous and \(M \subset X\) is a minimal set of \(f\) then \(M\) is either a finite set (a periodic orbit of \(f\)) or a Cantor set. This is in fact a characterization because also conversely, whenever \(M \subset X\) is a finite or a Cantor set then there is a continuous map \(f : X \to X\) such that \(M\) is a minimal set of \(f\ .\) Among one-dimensional spaces, the characterization of minimal sets is known for graphs --- minimal sets on connected graphs are characterized as finite sets, Cantor sets and unions of finitely many pairwise disjoint simple closed curves, see [BHS] or [Mai]. The full characterization of minimal sets on dendrites and on local dendrites can be found in [BDHSS].

With the exception of maps of zero and some one-dimensional spaces, the dynamics of arbitrary continuous maps is not extensively studied. This is quite understandable because continuity puts little restriction on maps of spaces of dimension higher than 1. In particular, in higher dimensions the topological structure of minimal sets is much more complicated and only few results and some important examples are known.

However, for some classes of maps which are special from the dynamical or topological point of view, the structure of minimal sets can be partially described regardless of the dimension of the phase space. One result of this kind is that if a dynamical system \((X,f)\) is topologically transitive then every minimal set of \(f\) is either nowhere dense or it is the whole space \(X\ .\) The same is true for homeomorphisms. In fact, if \((X,h)\) is a dynamical system and \(h\) is a homeomorphism then the boundary of a minimal set \(M\) is \(h\)-invariant (and closed), hence is equal to the set \(M\) or is empty. Thus, a minimal set of a homeomorphism either has empty interior (i.e., it is nowhere dense in \(X\)) or it is a clopen subset of \(X\ .\) Consequently, if \(X\) is connected, then the homeomorphism \(h\) has only nowhere dense minimal sets, with one possible exception when the whole space \(X\) is minimal for \(h\ .\)

On manifolds we know, due to [KST1], that if \(\mathcal{M}^2\) is a compact connected \(2\)-dimensional manifold, with or without boundary, \(f: \mathcal{M}^2\to \mathcal{M}^2\) is a continuous map and \(M \subset \mathcal{M}^2\) is a minimal set of the dynamical system \((\mathcal{M}^2,f)\) then either \(M = \mathcal{M}^2\) or \(M\) is a nowhere dense subset of \(\mathcal{M}^2\ .\) Moreover, by [BOT], the former case is possible only if \(\mathcal{M}^2\) is a torus or a Klein bottle. To find a full topological characterization of minimal sets on compact, connected \(2\)-manifolds is a very difficult task. Of course, some examples of `strange' minimal sets of continuous maps on \(2\)-manifolds are scattered in the literature (e.g., the Sierpiński curve on the \(2\)-torus, see [BKS], or a pseudocircle, see [Hand]). One can also think of embedding known one-dimensional minimal systems into a \(2\)-manifold. But all this is far from giving a characterization of minimal sets.

It is an open problem whether, for \(n > 2\ ,\) on compact connected \(n\)-dimensional manifolds proper minimal sets with nonempty interior exist.

Minimality and chaos

Figure 1: The relations between some definitions of chaos for minimal(!) dynamical systems (no other implications work except of those which can be deduced using the transitivity of implication, see e.g. [Kol]). For minimal systems the notions of weak mixing, scattering and generic chaos are equivalent.

The different notions of chaos at least share a common motivating intuition of unpredictability due to the divergence of nearby orbits. Different definitions begin with different interpretations of this divergence. We present here some popular ideas and relations between them for minimal dynamical systems (see Fig.1).

Let \((X,f)\) be a dynamical system. The idea of sensitivity was formalized in Auslander and Yorke [AY] and popularized in Devaney [Dev]. A point \(x\in X\) is called Lyapunov stable if, for any \(\varepsilon > 0\ ,\) there exists \(\delta > 0\) such that the inequality \(d(x,y)< \delta\) yields \(d (f^n(x),f^n(y)\leq \varepsilon\) for all integers \(n \geq 0\ .\) This condition means that the iteration sequence \(\{f^n: n\geq 0\}\) is equicontinuous at the point \(x\ .\) A point of this type is therefore also called an equicontinuity point. The system \((X,f)\) is called almost equicontinuous if there is a dense \(G_\delta\) set of equicontinuity points.

So, a point \(x\in X\) is not Lyapunov stable if there is \(\varepsilon > 0\) such that arbitrarily close to \(x\) there are points \(y\in X\) with \(d (f^n(x),f^n(y) > \varepsilon\) for some \(n > 0\ .\) We then say that \(x\) is Lyapunov \(\varepsilon\)-unstable. A system \((X, f)\) is said to exhibit sensitive dependence on initial conditions (or is briefly called sensitive) if there exists \(\varepsilon > 0\) such that every point \(x \in X\) is Lyapunov \(\varepsilon\)-unstable.

Studying maps of the interval, Li and Yorke [LY] suggested that the 'divergent pairs' to consider are the pairs \((x,y)\) which are proximal but not asymptotic. We will call a pair \((x,y) \in X \times X\) a Li-Yorke pair, or a scrambled pair, when \(\liminf_{n \rightarrow \infty} \rho(f^{n}(x),f^{n}(y)) = 0 \quad \mbox{ but } \quad \limsup_{n \rightarrow \infty} \rho(f^{n}(x),f^{n}(y)) > 0.\) A system is called Li-Yorke chaotic when it contains an uncountable scrambled set. A subset \(A \subset X\) is \(scrambled\) when any pair of distinct points in \(A\) is a Li-Yorke pair. A system is called generically chaotic when the set of all Li-Yorke pairs is a residual subset of \(X \times X\ .\)

The following concept from [AK] links the Li-Yorke versions of chaos with the notion of sensitivity to initial conditions. A dynamical system \((X,f)\) is called Li--Yorke sensitive if there exists a positive \(\delta\) such that for every \(x \in X\) and every neighborhood \(U_x\) of \(x\) there exists a point \( y\in U_x\) such that the pair \((x,y)\) is Li-Yorke with modulus \(\delta\) (i.e., in the definition of a Li-Yorke pair, \(\limsup_{n \rightarrow \infty} \rho(f^{n}(x),f^{n}(y))\) is greater than \(\delta\) rather than just positive).

For subsets \(A\) and \(B\) of \(X\) we define the hitting time set \(n(A,B) : = \{ n \geq 0 : A \cap T^{-n}B \not= \emptyset \}\ .\) Recall that \((X,f)\) is transitive if for every pair of non-empty open subsets \(U\) and \(V\) of \(X\) the hitting time set \(n(U,V)\) is non-empty, hence infinite. \((X,f)\) is (topologically) mixing if for every pair of non-empty open subsets \(U\) and \(V\) of \(X\) the hitting time set \(n(U,V)\) is co-finite. A system \((X,f)\) is called weakly mixing when the product system \((X \times X, f \times f)\) is transitive. The Furstenberg Intersection Lemma says that for weakly mixing systems the collection of sets \( \{ n(U,V) \cap [k,\infty) : U, V\) non-empty open in \(X\) and \(k \geq 0 \}\) generates a filter (see Akin [Ak], p. 88). A dynamical system is scattering if and only if its cartesian product with any minimal dynamical system is transitive.

Finally, a dynamical system is called \(K\)-mixing (or a topological \(K\)-system) if every nontrivial finite open cover (each element is not dense) has positive topological entropy. Minimal topological \(K\)-systems exhibit all kinds of chaos considered in Fig. 1. For more information on topological \(K\)-systems see e.g. [HY], [HSY] and references therein.

Topological transformation groups and minimality

A topological transformation group (abbreviation: ttg) is a triple \(\langle T,X,\pi\rangle\) where \(X\) is a Hausdorff topological space, \(T\) is a topological group and \(\pi : T\times X \to X\) is a jointly continuous action of \(T\) on \(X\ .\) A ttg with \(T=\mathbb R\) is called a continuous flow (and a ttg with \(T=\mathbb Z\ ,\) i.e. a ttg with discrete time, is sometimes called a discrete flow). The orbit of a point \(x\in X\) is the set \(\{\pi(t,x):\, t\in T\}\) and a set \(M\subseteq X\) is invariant if it contains orbits of all its points. Then the definitions of minimal sets and minimality for a ttg are analogous to those for maps. However, if \(T=\mathbb Z\) sometimes misunderstandings arise about the definition of minimality. To explain them, recall that if \(\langle T,X,\pi\rangle\) is a ttg and \(t\in T\) then the mapping \(\pi^t : x\mapsto \pi(t,x)\) is a continuous map which is in fact a homeomorphism of \(X\) onto \(X\ .\) It is called the \(t\)-transition, or time \(t\)-map, of the ttg. The map \(t\mapsto \pi^t\) is a homomorphism of the group \(T\) into the group of homeomorphisms of \(X\) onto \(X\ .\) Thus, to define a ttg \(\langle \mathbb T,X,\pi\rangle\) with \(T=\mathbb Z\) is the same as to choose a homeomorphism \(h = \pi^1\ .\) However, while in \(\langle \mathbb Z,X,\pi\rangle\) the orbit of a point \(x\in X\) is the full orbit \(\{h^n (x):\, n\in \mathbb Z\}\ ,\) in the dynamical system \((X,h)\) (when \(h\) is viewed as just a map) the orbit of \(x\) is the forward orbit \(\{h^n(x):\, n=0,1,2,\dots\}\ ,\) if not stated otherwise (see Section 1). For a homeomorphism, minimality in the sense of the density of all full orbits is in general not equivalent to minimality in the sense of the density of all forward orbits. There are locally compact (but not compact) metric spaces which admit minimal homeomorphisms in the former sense but do not admit any minimal map in the latter one. However, if \(X\) is a compact metric space, these two definitions are equivalent.

For the structure of minimal sets of a ttg the same alternative holds as the one discussed above in the case of a homeomorphism -- a minimal set is either nowhere dense or clopen. Hence, if a minimal set of a ttg has nonempty interior then it is a union of components of \(X\) (just one component if the group \(T\) is connected) and so it coincides with \(X\) provided \(X\) is connected.

The problem of which topological properties characterize a space \(X\) that is a phase space of some minimal ttg, is far from being solved even for groups \(\mathbb R\) and \(\mathbb Z\ .\) For \(T=\mathbb Z\) and compact \(X\ ,\) see the pieces of information on minimal homeomorphisms in previous sections.

For connected groups the following general result holds. If \(X\) is a finite-dimensional compact metric space, \(T\) a connected group and \(\langle T,X,\pi\rangle\) a minimal ttg, then \(X\) is a Cantor manifold. (A compact metric space \(X\) with \(\dim X = n\) is called a Cantor manifold if \(X\) cannot be presented as a union of two nonempty closed subsets \(X_1\) and \(X_2\) with \(\dim (X_1\cap X_2) \leq n-2\ .\)) In particular, if \(X\) is a compact connected \(2\)-manifold and \(T=\mathbb R\) then \(X\) is necessarily a torus. The still unproved Gottschalk's conjecture says that there is no continuous flow on the \(3\)-dimensional sphere.

For more information on minimality in the setting of topological transformation groups see the books [GH], [Ell1], [Br], [Au] and [Vri].

Concerning the connection between minimal continuous flows and minimal homeomorphisms, if a compact metric space \(X\) admits a minimal continuous flow then for residually many \(t\in\mathbb R\) the time \(t\)-map of the flow is minimal and so \(X\) admits also a minimal homeomorphism, see [Fa]. The converse is not true (the Klein bottle does not admit a minimal continuous flow though it admits a minimal homeomorphism).

On structure theorems for minimal flows

In this section a topological transformation group \(\langle T, X, \pi \rangle\) will be simply called a flow and denoted by \((X,T)\ .\) Moreover, instead of \(\pi^t(x)=\pi(t,x)\) we will just write \(tx\ .\) We will further assume that the phase spaces of all considered flows are compact Hausdorff spaces, while \(T\) will be any (fixed) topological group. In this section we partially follow the article Topological dynamics.

If the transition homeomorphisms \(\pi^t\) defined by the elements of \(T\) form an equicontinuous family then the flow is called equicontinuous. If \(X\) is a metric space with metric \(d\ ,\) this means that given \(\varepsilon >0\) there is a \(\delta >0\) such that if \(d(x,y) < \delta\) then \(d(tx, ty) < \varepsilon\) for all \(t\in T\ .\) If metrizability is not assumed then the definition uses the unique compatible uniformity.

A pair of points \((x,y) \in X\times X\) is called proximal if, in the metric case, for every \(\varepsilon >0\) there is \(t\in T\) with \(d(tx, ty) < \varepsilon\ .\) Again, in the general case the uniformity is used in the definition. A pair of points is called distal if it is not proximal. A flow is called distal if all pairs \((x,y)\) with \(x\neq y\) are distal. An equicontinuous flow on a compact Hausdorff space is distal, but the converse is not true in general. If \((X,T)\) and \((Y,T)\) are flows and \(p: X\to Y\) a continuous surjective map such that \(p(tx) = tp(x)\) for every \(t\in T\) and every \(x\in X\ ,\) we say that the flow \((Y,T)\) is a factor of the flow \((X,T)\ ,\) or that \((X,T)\) is an extension of \((Y,T)\ .\) The map \(p\) is called a homomorphism of the flows or an extension or a factor (map). The extension \(p\) is called proximal whenever every pair \((x_1,x_2)\) of points in \(X\) with \(p(x_1) = p(x_2)\) is proximal in \((X,T)\ .\) Similarly, \(p\) is called a distal extension whenever for all \(x_1 \neq x_2\) with \(p(x_1) = p(x_2)\) the pair \((x_1,x_2)\) is distal. The extension \(p\) is called equicontinuous if, in the metric case, for every \(\varepsilon>0\) there exists a \(\delta>0\) such that \(d(tx_1,tx_2)<\varepsilon\) for all \(t\in T\) and for all \(x_1,x_2\) in \(X\) with \(p(x_1)=p(x_2)\) and \(d(x_1,x_2)<\delta\ .\) If an extension is equicontinuous then it is distal; the converse is not true in general.

If a flow is obtained by an equicontinuous extension of an equicontinuous flow then it need not be equicontinuous but it is necessarily distal. In fact more is true: An equicontinuous extension of a distal flow (and even a distal extension of a distal flow) is distal. Therefore if we start with a distal flow, say the trivial one point flow, and extend it equicontinuously again and again, possibly transfinitely many times by passing to inverse limits of flows at limit ordinals (see e.g. [Au] or [Vri] for the definition of the inverse limit of flows), we will be always in the class of distal flows. The following deep Furstenberg structure theorem says that the converse is also true:

Let \((X,T)\) be a distal minimal flow on a compact Hausdorff space. Then there is an ordinal number \(\eta\) and a family of minimal flows \((X_{\alpha},T)\) for \(\alpha \leq \eta\) such that \((X_0,T)\) is the trivial one point flow, for \(\alpha < \eta\) the flow \((X_{\alpha +1},T)\) is an equicontinuous extension of \((X_{\alpha},T)\ ,\) for any limit ordinal \(\alpha \leq \eta\) the flow \((X_{\alpha},T)\) is the inverse limit of the flows \((X_{\beta},T)\ ,\) \(\beta <\alpha\ ,\) and finally \((X_{\eta},T)=(X,T)\ .\)

As a corollary of this theorem we get that a non-trivial distal minimal flow on a compact Hausdorff space always has a non-trivial equicontinuous factor (the flow \((X_1,T)\)).

The Furstenberg structure theorem was extended by several authors. In particular, there are structure theorems for so called point distal minimal flows, prodal minimal flows, normal minimal flows. There is also a structure theorem for general minimal flows (equicontinuous, proximal and so-called weakly mixing extensions appear in it). For more details see [Gl], [Au], [Vri] and references which can be found there.

The financial support from VEGA, grant 1/0855/08 is highly appreciated.


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Internal references

  • Tomasz Downarowicz (2007) Entropy. Scholarpedia, 2(11):3901.
  • Brian Marcus and Susan Williams (2008) Symbolic dynamics. Scholarpedia, 3(11):2923.
  • Roy Adler, Tomasz Downarowicz, Michał Misiurewicz (2008) Topological entropy. Scholarpedia, 3(2):2200.

See also

Chaos, Entropy, Entropy in Chaotic Dynamics, Ergodic theory, Topological dynamics, Topological entropy, Topological transitivity.

External links


Wikipedia: Chaos theory

Wikipedia: Topological entropy

Wikipedia: Topological entropy (in physics)

PlanetMath: Topological entropy

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