# Topological Transitivity

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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 $$F_\sigma\ .$$

## Hitting times and notions related to transitivity

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

• 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.
• Roy Adler, Tomasz Downarowicz, Michał Misiurewicz (2008) Topological entropy. Scholarpedia, 3(2):2200.