There are various types of transitivity. Let be a compact metric space and a continuous map on . There are several definitions of transitivity.

(Topological Transitivity). For every pair of nonempty open sets and in , there is a positive integer such that ,

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

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

(W-Transitivity) The *whole* orbit is dense in .

(F-Transitivity) The *forward* orbit is dense in .

(B-Transitivity) The *backward* orbit is dense in .

Example 1. with discrete topology, . Then is PT but not TT.

Example 2. . Let be the set of periodic points of with induced topology. Then is TT, but not PT.

Example 3. be the one-point compactification of . Let and . Then is WT, but not FT or BT.

Proposition 1. If has isolated points and the homeomorphism is transitive, then is of a single orbit of .

Proposition 2. For continuous maps

‘PT implies TT’ if has no isolated point.

‘TT implies PT’ if 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 and . Let be the set of points whose whole orbit is dense. Similarly we define and .

Proposition 3. is -saturated and is -saturated. Note that may not be bi-saturated.

In general all these are totally different. Take the full shift for example (hence for all horseshoes and hyperbolic systems). If we arrange the whole word one by one to generate a and assign arbitrary values for (for example for all ), then the resulting point is a FT but not BT. Similarly there is some point that is BT but not FT.

Question 1: Is

Answer: Yes if is a compact Baire space.

Proof. Let be a homeomorphism on and . Then every open invariant set is dense in since it contains . On the other hand we have . By closeness, one of them, say , has nonempty interior. Then contains an open dense subset of and hence by the closeness of . That is .

Question 2: what about the set ?

Answer: It is residual.

Let be a compact Baire space. Assume that . Then a more involve argument shows that both are nonempty and automatically dense subset. In particular, is also dense .

Question 3: what is the difference of ?

Answer: It is of zero probability. That is, for each .

Let . Then for each open set , . Let be a subbasis of the topology on . Then .