Topological transitivity
The concept of topological transitivity goes back to G. D. Birkhoff [1] who introduced it in 1920 (for flows). This article will concentrate on topological transitivity of dynamical systems given by continuous mappings in metric spaces. Intuitively, a topologically transitive dynamical system has points which eventually move under iteration from one arbitrarily small open set to any other. Consequently, such a dynamical system cannot be decomposed into two disjoint sets with nonempty interiors which do not interact under the transformation.
Topological transitivity versus the existence of a dense orbit
Let \(X\) be a metric space with the metric \(d\) and \(f: X\to X\) be continuous. The dynamical system \((X,f)\) is called topologically transitive if it satisfies the following condition.
- (TT) For every pair of non-empty open sets \(U\) and \(V\) in \(X,\) there is a non-negative integer \(n\) such that \(f^n(U)\cap V \neq \emptyset .\)
However, some authors choose, instead of (TT), the following condition as the definition of topological transitivity.
- (DO) There is a point \(x_0\in X\) such that the orbit \(\{x_0, f(x_0), \dots f^n(x_0), \dots \}\) is dense in \(X .\)
Unfortunately, the two conditions are independent in general. To see that (DO) does not imply (TT), consider \(X=\{0\}\cup \{{1/n}\,:\,n\in \Bbb N \}\) endowed with the usual metric and \(f: X \to X\) defined by \(f(0)=0\) and \(f({1/n})={1/(n+1)}, n=1,2,\dots\ .\) To show that (TT) does not imply (DO), start with \(I=[0,1]\) and the standard tent map \(g(x)=1-|2x-1|\) from \(I\) to itself. Then let \(X\) be the union of all periodic orbits of \(g\) and \(f=g_{|X} .\) The system \((X,f)\) does not satisfy the condition (DO), since \(X\) is infinite (dense in \(I\)) while the orbit of any periodic point is finite. On the other hand, every pair of subintervals of \(I\) shares a periodic orbit of the tent map and so \((X,f)\) satisfies (TT).
Nevertheless, under some additional assumptions on the phase space (or on the map) the two conditions (TT) and (DO) are equivalent. In fact, if \(X\) has no isolated point then (DO) implies (TT) and if \(X\) is separable and second category then (TT) implies (DO) ([Silv]).
The systems satisfying (DO) are sometimes called point transitive. In the sequel, when speaking on transitivity, we have topological transitivity in mind.
Some of the equivalent definitions of transitivity
For subsets \(A\) and \(B\) of \(X\) define the hitting time set \(n(A, B)=\{n\geq 0: A\cap f^{-n}(B)\neq \emptyset \}\ .\) Let \((X,f)\) be a dynamical system. Then the following are equivalent:
- \(f\) is topologically transitive (i.e., (TT) is fulfilled),
- for every pair of non-empty open sets \(U\) and \(V\) in \(X\) the hitting time set \(n(U, V )\) is infinite,
- for every non-empty open set \(U\) in \(X,\) the (forward) orbit of \(U\) is dense in \(X\ ,\)
- for every non-empty open set \(U\) in \(X,\) the backward orbit of \(U\) is dense in \(X\ ,\)
- every proper closed (forward) invariant subset of \(X\) is nowhere dense,
- every backward invariant subset of \(X\) with non-empty interior is dense.
When studying topological transitivity, it is not restrictive to consider only phase spaces without isolated points. In fact, if a transitive system has an isolated point then the system is trivial, consisting of just one periodic orbit.
Some examples of transitive maps
Every minimal dynamical system is transitive (a system is minimal if all orbits are dense).
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 [R] 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 \((i_n)_{n>0}\) for which \(0\leq i_n \leq k_n\ .\) 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.
Non-minimal transitive systems.
Example 3. The logistic map \(g(x)=4x(1-x)\) defined on the interval \([0,1]\) is topologically transitive. This follows from the fact that the tent map \(f(x)=1-|2x-1|\) is topologically conjugate to \(g\ ,\) the conjugating homeomorphism being \(h(x)= \sin ^2 (\pi x/2)\ .\)
Example 4. Symbolic dynamics provides many examples of transitive dynamical systems. A subshift of finite type is topologically transitive if and only if its transition matrix \(M\) is irreducible which means that for every \((i,j)\) there is a positive integer \(n\) such that the \((i,j)^{th}\) entry of the matrix power \(M^n\) is strictly positive.
Every compact, connected \(n\)-manifold (possibly with boundary), \(n\geq 2\ ,\) admits a transitive homeomorphism [Al]. Moreover, such homeomorphisms are typical, with respect to the uniform topology, in the space of all measure preserving homeomorphisms [DF]. The first results of this kind were [Ox] and [OU] (see also [AnK] and [AP]).
Concerning noncompact manifolds, Besicovitch [Bes] gave the first explicit example of a transitive homeomorphism of the plane. Transitive homeomorphisms exist in fact on \(\sigma\)-compact connected \(n\)-manifolds, \(n\geq 2\ ,\) and in some cases they are dense, in the compact open topology, in the space of all measure preserving homeomorphisms [AP1]. See also [AP].
There are spaces, even metric continua, which do not admit transitive maps. Though for instance a characterization of locally compact subspaces of the real line admitting a transitive map is known (see [NK]), no characterization of compact metric spaces admitting transitive maps is known. However, in [AC] it is proved that every finite union of disjoint nondegenerate Peano continua in \(\mathbb R^n\) admits a transitive map.
Transitivity of a map and its iterates
A system \((X,f)\) is called totally transitive, if the system \((X,f^n)\) is transitive for every \(n\geq 1\ .\) If \((X,f)\) is topologically transitive but \((X,f^n)\) is not, then there are an integer \(k\geq 2\) dividing \(n\) and sets \(X_0, X_1, ..., X_{k-1}\) such that:
- \(X= X_0\cup X_1\cup ... \cup X_{k-1}\ ,\)
- each \(X_i\) is regular closed (i.e., it is the closure of its interior),
- \(X_i\cap X_j\) is nowhere dense whenever \(i\not = j\ ,\)
- \(f(X_i)\subseteq X_{i+1 (\mod k)}\ ,\)
- \(f^n\) (hence also \(f^k\)) is transitive on each \(X_i\ .\)
For more details on this topic see [Ban].
Transitive and intransitive points
Any point with dense orbit is called a transitive point. A point which is not transitive is called intransitive. The image of an intransitive point is intransitive and if the phase space has no isolated point then also the image of a transitive point is transitive. Further, if the phase space of a system \((X,f)\) has a countable base \(\{U_i\}_{i=1}^{\infty}\) of open sets then the set of transitive points can be expressed in the form \(\bigcap _{k=1}^{\infty} \left ( \bigcup _{n=0}^{\infty} f^{-n}(U_k)\right )\) and so it is a \(G_\delta\) set.
The set of transitive or intransitive points of \((X,f)\) will be denoted by \(tr (f)\) or \(intr (f)\ ,\) respectively. Assume that \(X\) is a compact metric space without isolated points. Then one of the following holds (see [Kin] or [KS]):
- \(tr (f) = \emptyset\) and \(intr (f) = X,\)
- \(tr (f) = X\) and \(intr (f) = \emptyset\) (in this case the system \((X,f)\) is called minimal),
- \(tr (f)\) is dense \(G_\delta\) and \(intr (f)\) is dense \(F_\sigma\ .\)
A system \((X, f)\) is called topologically weakly mixing when the product system \((X\times X, f\times f)\) is topologically transitive. An equivalent definition is that for every pair of non-empty open subsets \(U\) and \(V\) of \(X\) the hitting time set \(n(U, V )\) contains arbitrarily long intervals of positive integers ([Fur], see also [Ak1]). Some stronger notions of mixing in topological dynamics can also be characterized in terms of hitting time sets (see [GW]).
Transitivity and chaos
Transitivity is a widely accepted feature of chaos. It is often required in definitions of chaos as one of several ingredients. However, transitivity alone is pretty compatible with a very regular behavior of trajectories --- in fact, a transitive system is either very regular or very non-regular. To describe this dichotomy in more details, we need to recall some definitions.
A point \(x\in X\) is called Lyapunov stable if, for any \(\epsilon > 0\ ,\) there exists \(\delta > 0\) such that the inequality \(d(x,y)< \delta\) yields \(d (f^n(x),f^n(y)\leq \epsilon\) for all integers \(n > 0\ .\) 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 \(\epsilon > 0\) such that arbitrarily close to \(x\) there are points \(y\in X\) with \(d (f^n(x),f^n(y) > \epsilon\) for some \(n > 0\ .\) We then say that \(x\) is Lyapunov \(\epsilon\)-unstable. A system \((X, f)\) is said to exhibit sensitive dependence on initial conditions (or is shortly called sensitive) if there exists \(\epsilon > 0\) such that every point \(x \in X\) is Lyapunov \(\epsilon\)-unstable. The notion of sensitivity for a system \((X, f)\) is equivalent to the condition that there exists \(\epsilon > 0\) such that for every non-empty open set \(U\) in \(X\) there exist \(x, y \in U\) with \(\limsup_{n\to \infty} d(f^n(x), f^n(y))> \epsilon\ .\)
A transitive system \((X,f)\) is either sensitive or almost equicontinuous. In the latter case the set of equicontinuity points coincides with the set of transitive points and the map \(f\) is a homeomorphism and is uniformly rigid (see [AAB]). For more information on the relation between transitivity and various concepts of chaos see [AK], [Bl], [Kol], [XJ].
Transitivity and topological entropy
In compact metric spaces in general there is no connection between topological transitivity and topological entropy. A system with positive topological entropy need not of course be transitive (transitivity is a global property of a system while a system may have positive topological entropy on some small invariant subset, without being transitive). On the other hand, a transitive system may have zero topological entropy. However, in some spaces topologically transitive maps have necessarily positive topological entropy. For instance, on a real compact interval every transitive map has topological entropy at least \((1/2)\log 2\) and there is a transitive map with topological entropy equal \((1/2)\log 2\ .\) For more information on best lower bounds for the topological entropy of transitive maps in various spaces see [AKLS], [BS].
Transitivity of (semi)group actions
Topological transitivity can be studied in a more general setting. An action of a semigroup \(G\) on a space \(X\) is called topologically transitive if, for every pair of non-empty open sets \(U\) and \(V\) in \(X\ ,\) there is an element \(g\in G\) such that \(g(U) \cap V \neq \emptyset\ .\) Usually it is at least assumed that the action is separately continuous in the space variable which means that, for each element \(g\in G\ ,\) the corresponding map \(g: X\to X\) is continuous. When \(G\) is a group, usually it is required that for each element \(g\in G\ ,\) the corresponding map \(g: X\to X\) be a homeomorphism, see e.g. [CKN].
However, note that the required partial continuity of an action of a (semi)group is just the minimum requirement in topological dynamics. In fact, abstract topological dynamics is usually developed in the context of (semi)flows. A (semi)flow is a jointly continuous action of a topological (semi)group \(G\) on a space \(X\ .\)
In this article we have in fact considered topological transitivity of \(\mathbb Z^{+}\)-actions where \(\mathbb Z^{+}\) is the additive semigroup \(\{0,1,2,\dots\}\ .\) Such an action is given by a map \(f: X\to X\ .\) We have assumed continuity of \(f\) and so our dynamical system was a particular case of a semiflow (\(\mathbb Z^{+}\) is a topological semigroup with respect to the discrete topology).
For more information on topological transitivity of flows see [GH] and [dV] (in [GH] it is called regional transitivity and in [dV] it is called topological ergodicity). Topological transitivity of partially continuous actions of groups is discussed in [CKN].
The financial support from VEGA, grant 1/0855/08 is highly appreciated.
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Internal references
- Olaf Sporns (2007) Complexity. Scholarpedia, 2(10):1623.
- Yuri A. Kuznetsov (2007) Conjugate maps. Scholarpedia, 2(12):5420.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Tomasz Downarowicz (2007) Entropy. Scholarpedia, 2(11):3901.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
- David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.
- Brian Marcus and Susan Williams (2008) Symbolic dynamics. Scholarpedia, 3(11):2923.
- Joseph Auslander (2008) Topological dynamics. Scholarpedia, 3(6):3449.
- Roy Adler, Tomasz Downarowicz, Michał Misiurewicz (2008) Topological entropy. Scholarpedia, 3(2):2200.
See also
Chaos, Complexity, Entropy, Entropy in Chaotic Dynamics, Ergodic theory, Topological dynamics, Topological entropy.
External links
Wikipedia
Wikipedia: Topological entropy