Introduction to topological dynamical systems

For topological dynamical systems, there is a topological structure on the phase space. In practice we hardly care about the first finite items in orbit, the concept of positive half-orbits defined by pure algebra and the negative half-orbit of reversible mappings let's be located in $\omega $ limit set and $\alpha $ limit set: (assumed to be discrete systems)

\[\omega \left (x \right) = \bigcap\limits_{n \in \mathbb{n}} {\overline {\bigcup\limits_{t \geqslant n} {{f^t}\left (x \ Right)}}} \]

\[\alpha \left (x \right) = \bigcap\limits_{n \in \mathbb{n}} {\overline {\bigcup\limits_{t \geqslant n} {{F^{-t}}\left (x \right)}}}\]

Plainly, the whole of the points that can be approximated with the sub-columns of the positive half-rails is the $\omega $ limit set, which is

\[\omega (x) = \{y\in x | \exists n_i \to \infty \text{with} T^{n_i} (x) \to y\}\]

Next, we introduce some concepts:

If a point is in its own $\omega $ limit set, it is called a recovery point (in topological sense).

If any neighborhood of a point $U $, you can always find a $n $ makes ${t^{-n}}\left (u \right) \cap u \ne \varnothing$, which is called a non-loitering point. As you can see later, the complexity of system topological entropy is completely concentrated on the non-wandering set. For convenience, for $U, V\subset x$, will $N (u,v) =\{n\in \mathbb{z}_{+}: U\cap t^{-n}v \ne \varnothing \}$ is called the reply time set. In other words, the non-loitering point is the point where any neighborhood $U $, the reply time set $N (u,u) $ are not empty.

Introduction to topological dynamical systems