Chain recurrence

Remarks

For a continuous flow $\varphi_t$, an $(\varepsilon, T)$-chain from $x$ to $y$ is a sequence $x = x_0, \ldots, x_n = y$ with times $t_1, \ldots, t_n \ge T$. It must satisfy $d(\varphi_{t_i}(x_{i-1}), x_i) < \varepsilon$ for each $i$. A point $x$ is chain recurrent for the flow if such chains from $x$ to $x$ exist for all $\varepsilon > 0$ and $T > 0$. The set $\mathcal{R}(f)$ is closed and contains the non-wandering set $\Omega(f)$. Example: for the identity map, $x_0 = x_1 = x$ is an $\varepsilon$-chain from $x$ to $x$ for every $\varepsilon$, so every point is chain recurrent. Non-example: let $f(x) = x/2$ on $[0, 1]$ and $x > 0$, $0 < \varepsilon < x/2$. If $x_{i-1} \le x$, then $x_i < x_{i-1}/2 + \varepsilon < x$, so by induction every $x_i$ with $i \ge 1$ is less than $x$; no $\varepsilon$-chain returns to $x$. For this map $\mathcal{R}(f) = \set{0}$.