Conley Theorem (or fundamental theorem of dynamical systems)

Question

Notations:

• $\mathcal{R}(f)$ denotes the chain recurrent set of $f$
• $NW(f)$ denotes the non wandering set of $f$
• $R(f)$ denotes the recurrent set of $f$ ($x: x\in \omega(x)$)

Given compact metric space X and homeomorphism $f: X \rightarrow X$, $g: X \rightarrow \mathbb{R}$ is a complete Lyapunov function for $f$ if:

$\forall \ p \notin \mathcal{R}(f)$, $\ g(f(p)) < g(p)$

$\forall \ p, q \in \mathcal{R}(f)$, $\ g(p) = g(q)$ iff $ p \sim q$

$g(\mathcal{R}(f))$ is compact and nowhere dense in $ \mathbb{R}$

Conley's Theorem (or fundamental theorem of dynamical systems):

Complete Lyapunov function exists for any homeomorphisms on compact metric spaces.

My Question: I'd like to understand why this theorem is called fundemental theorem of dynamical systems, It seems to me it says that the whole interesting dynamic $ f $ is contained in the chain recurrent set of $f.$ this was what I was told, but it is not very clear to me.

An example to explain my question: In the case of all non-wandering of $ f $ we prove that given any neighborhood $U$ of $NW(f)$ there is a uniform time $T$ for which $$ \text{Card} \{ k: f^k(x)\notin U \} \leq T $$

In this sense it seems clear that the non wandering set contains much of the interesting dynamics of $ f $.

However in the case of the chain recurrent set is not clear to me nothing that occurs in the previous example.

So like an explanation of why the whole interesting dynamic $ f $ is concentrated in the chain recurrent, and wanted to better understand why the theorem of Conley is called "fundamental".

Another question: Would an example of a dynamic system where $NW(f)\setminus \overline{R(f)}$ is not empty, or at least the idea of what kind of dynamic phenomenon can produce these point.

Answer 1

There is a classical example with $NW(f)\backslash \overline{R(f)}\neq\emptyset$, the so called Bowen's eye-like attractor (see the paper by Baladi, Bonatti and Bernard: Abnormal Escape Rates from Nonuniformly Hyperbolic Sets):

Every point on the dark curve is nonwandering, but only the two corners are recurrent (in fact these two are fixed).

Why is the theorem called Fundamental? One reason is that it is the 'correct' setting of $C^1$ stability conjecture (see the discuss here).

Theorem: Let $f$ be a diffeomorphism on a closed manifold $M$. Then the following are equivalent:
1. the map $f$ is structurally stable;
2. $\mathcal{R}(f)$ is hyperbolic;
3. $f$ is $\mathcal{R}$-stable.

This modern version is very succinctly comparing to the version using the nonwandering set, which involves with no-cycle condition and transversality condition.

Also I copied a few words from the paper mentioned in Barry's comment:

'The theorem is fundamental in the sense that it deals with the basic question of the field. It is also fundamental in that it encompasses such big ideas in such a small, concise statement.'

Compared to Fundamental Theorem of Arithmetic and Fundamental Theorem of Algebra, Norton wrote:

'the space on which the dynamics take place, can be decomposed uniquely into its basic dynamical parts: points whose dynamics can be described as exhibiting a particular type of recurrence, and points which proceed in a gradient-like fashion.'