Try searching for topics concerning "Learning Dynamical Systems" and "Predictive State Representation." Here is a possible reference.

I'm not sure Ergodicity in full generality is useful. If you're dealing with a Markov chain with a stationary distribution then there will be some ergodicity involved concerning the visitation of the steady states. Also, there's something called "Ergodic Time Series" which might be interesting to you.

First let me address what your various textbooks say.

Given a manifold $M$, to say that a vector field $f$ is a mapping from $M$ to $TM$ is incomplete. Let me use $T_p M$ to denote the fiber of $TM$ over the point $p \in M$, in other words $T_p M$ is the tangent space of $M$ at $p$. A vector field on a manifold $M$ is *not* just any old mapping from $M$ to $TM$. Instead, a vector field on $M$ is a mapping $f : M \to TM$ such that for each $p \in M$ we have $f(p) \in T_p M$, in other words $f(p)$ is a vector in the tangent space at $p$.

In the special case where $M = \mathbb{R}^n$, if you keep all of this notation in mind, then your two textbooks are saying essentially the same thing. The tangent bundle in this case is a product $T \mathbb{R}^n = \mathbb{R}^n \times \mathbb{R}^n$, and the tangent space at each point $p\in \mathbb{R}^n$ has the form
$$T_p \mathbb{R}^n = \{(p,v) \,|\, v \in \mathbb{R}^n\}
$$
where the vector operations on the vector space $T_p \mathbb{R}^n$ are defined by simply ignoring the $p$ coordinate, i.e. $(p,v) + (p,w) = (p,v+w)$ and similarly for scalar multiplication. Because of this, there is a canonical isomorphism between vector fields expressed as functions
$$f : \mathbb{R}^n \to \mathbb{R}^n
$$
and vector fields expressed as functions
$$g : \mathbb{R}^n \to T \mathbb{R}^n
$$
This canonical isomorphism is given by the formula $g(p)=(p,f(p))$.

Regarding your last sentence, perhaps there may be elementary expositions of dynamical systems that restrict attention to $M=\mathbb{R}^n$, but the full theory of dynamical systems considers manifolds in all their full and general glory, and in this theory it is *not* sufficient to consider $\mathbb{R}^n$. Dynamical systems on spheres, on toruses, and on all kinds of manifolds are important.

I would also point out that it is misleading to say that a dynamical system on a manifold $M$ is a vector field. What is important in dynamical systems is not the vector field in particular, but its integral curves and their behavior over long time spans.

## Best Answer

In general any recurrent point is a non-wandering point, but not vice versa.

For an example consider the expanding map $f:\mathbb{T}\to \mathbb{T}, x\mapsto 2x$ and the point $x_0=\dfrac{1}{2}$. Then $f^n(x_0)=0$ for $n\in\mathbb{Z}_{\geq1}$, so that $x_0$ is not an $f$-recurrent point (the orbit of $x_0$ does not approximate $x_0$ arbitrarily well).

However it is $f$-non-wandering, since if $U\ni x_0$ is an open arc containing $x_0$, $f^n(U)$ will have length $2^n$ times the length of $U$, so that for some $n\in\mathbb{Z}_{\geq1}$, $f^n(U)$ will cover the whole circle; in particular $f^n(U)\cap U\neq \emptyset$.