Suspension flow

Remarks

The flow is well defined because the generating relation commutes with the translations $(x, s) \mapsto (x, s + t)$. Every class has a representative $(x, s)$ with $0 \le s < r(x)$. So $X_r$ can be pictured as the region under the graph of $r$, with the top point $(x, r(x))$ glued to the bottom point $(T(x), 0)$. For $0 \le s < r(x)$ and $t \ge 0$, the point $\varphi_t([x, s])$ equals $[T^n(x), s + t - \sum_{k=0}^{n-1} r(T^k(x))]$, where $T^k$ is the $k$-th iterate and $n \ge 0$ is the number of times the orbit has hit the roof. The first-return map of the flow to the base $\set{[x, 0] : x \in X}$ is $T$, and the return time of $[x, 0]$ is $r(x)$. If $\mu$ is a $T$-invariant measure, the product of $\mu$ with Lebesgue measure on the region under the graph of $r$ is invariant under the flow.

Example: let $X = \R / \Z$, $T(x) = x + \alpha \bmod 1$ and $r = 1$. The map $[x, s] \mapsto (x + \alpha s, s) \bmod \Z^2$ is well defined, since $(x, s + 1)$ and $(x + \alpha, s)$ both go to $(x + \alpha s + \alpha, s) \bmod \Z^2$. It identifies $X_r$ with the torus $\R^2 / \Z^2$, and it carries the suspension flow to the linear flow $(u, v) \mapsto (u + \alpha t, v + t)$ in the direction $(\alpha, 1)$.

The condition $\inf_X r > 0$ makes the sums $\sum_{k=0}^{n-1} r(T^k(x))$ and $\sum_{k=1}^{n} r(T^{-k}(x))$ tend to infinity, which is what gives every class a representative under the roof. Non-example: the doubling map $x \mapsto 2x \bmod 1$ is not a homeomorphism, so the construction above does not apply to it; for non-invertible maps one uses a suspension semiflow, defined only for $t \ge 0$. With $r = 1$, the space $X_r$ is the mapping torus of $T$.