topological transitivity

There are various types of transitivity. Let X be a compact metric space and f:X\to X a continuous map on X. There are several definitions of transitivity.

(Topological Transitivity). For every pair of nonempty open sets U and V in X, there is a positive integer n such that f^n(U)\cap V\neq\emptyset,

(Point Transitivity). There is a point x\in X such that the orbit of x is dense in X.

Since we do not assume that f is invertible, above only make sense for positive orbits. However if f is a homeomorphism, the second can be subdivided into three cases:

(W-Transitivity) The whole orbit \mathcal{O}_{\mathbb{Z}}(x) is dense in X.

(F-Transitivity) The forward orbit \mathcal{O}_{\mathbb{N}_0}(x) is dense in X.

(B-Transitivity) The backward orbit \mathcal{O}_{-\mathbb{N}_0}(x) is dense in X.

Example 1. X=\{a,b\} with discrete topology, f(a)=f(b)=b. Then (X,f) is PT but not TT.

Example 2. f:\mathbb{T}\to\mathbb{T},x\mapsto 2x. Let X be the set of periodic points of f with induced topology. Then (X,f) is TT, but not PT.

Example 3. X=\mathbb{Z}\cup\{\infty\} be the one-point compactification of \mathbb{Z}. Let f(n)=n+1 and f(\infty)=\infty. Then (X,f) is WT, but not FT or BT.

Proposition 1. If X has isolated points and the homeomorphism f:X\to X is transitive, then X is of a single orbit of f.

Proposition 2. For continuous maps

‘PT implies TT’ if X has no isolated point.
‘TT implies PT’ if X is separable and second category.

For good spaces (for example, a connected closed manifold) the two definitions PT and TT are equivalent. For now on we focus on the good spaces and try to distinguish the three subcategories of PT: WT, FT and BT.

Let W^s(x)=\{y\in X: d(f^nx,f^ny)\to 0, n\to+\infty\} and W^u(x)=\{y\in X: d(f^nx,f^ny)\to 0, n\to-\infty\}. Let WT be the set of points whose whole orbit is dense. Similarly we define FT and ~BT.

Proposition 3. FT is W^s-saturated and ~BT is W^u-saturated. Note that WT may not be bi-saturated.

In general all these are totally different. Take the full shift (\Sigma_2,\sigma) for example (hence for all horseshoes and hyperbolic systems). If we arrange the whole word \mathcal{W}=\bigcup_{n\ge1}\{0,1\}^n one by one to generate a (i_n)_{n\ge0} and assign arbitrary values for (i_n)_{n<0} (for example i_n=0 for all n<0), then the resulting point is a FT but not BT. Similarly there is some point that is BT but not FT.

Question 1: Is WT=FT\cup BT?

Answer: Yes if X is a compact Baire space.

Proof. Let f be a homeomorphism on X and x\in WT\neq\emptyset. Then every open invariant set is dense in X since it contains \mathcal{O}_{\mathbb{Z}}(x). On the other hand we have X=\omega(x)\cup\alpha(x). By closeness, one of them, say \omega(x), has nonempty interior. Then \omega(x) contains an open dense subset of X and hence \omega(x)=X by the closeness of \omega(x). That is x\in FT.

Question 2: what about the set FT\cap BT?

Answer: It is residual.
Let X be a compact Baire space. Assume that WT\neq\emptyset. Then a more involve argument shows that both FT, BT are nonempty and automatically dense G_\delta subset. In particular, FT\cap BT is also dense G_\delta.

Question 3: what is the difference of WT\backslash FT?

Answer: It is of zero probability. That is, \mu(WT\backslash FT)=0 for each \mu\in\mathcal{M}(f).
Let \mu\in\mathcal{M}(f). Then for each open set U, \mu(\bigcup_{k\in\mathbb{Z}}f^kU\backslash \bigcup_{k\ge0}f^kU)=0. Let \mathcal{U}=\{U_n\} be a subbasis of the topology on X. Then \mu(WT\backslash FT)\le\sum_{n}\mu(\bigcup_{k\in\mathbb{Z}}f^kU_n\backslash \bigcup_{k\ge0}f^kU_n)=0.

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