First-return map - Knowlpedia

For a transformation $T:X\to X$ and a subset $E$ , define the first-return time

r_E(x)=\min\{n\geq1:T^nx\in E\}

where this set is nonempty. The first-return map is $T_E(x)=T^{r_E(x)}x$ . In a probability-preserving system it is defined for almost every $x\in E$ , by recurrence.

Flows and sections

For a smooth flow and a transverse section, a Poincaré return map records the next intersection at positive time wherever a first such intersection exists. The flow's invariant volume need not restrict to a useful measure on a lower-dimensional section; an invariant transverse measure is a separate construction.

For the flow $(x,y)\mapsto(x+t,y+\theta t)$ on $\mathbb T^2$ , the section $x=0$ has constant return time one and return map $y\mapsto y+\theta$ .