Dynamical systems with cross-sections

Abstract:

The problem of classifying dynamical systems (flows) with global cross-sections in terms of the associated diffeomorphisms of the cross-sections is considered. Suppose that, for $i = 1,2,{\phi _i}$ is a ${C^r}$ flow $(r \geqslant 0)$ on the ${C^r}$ manifold ${M_i}$ that admits a global cross-section ${S_i} \subseteq {M_i}$ with associated diffeomorphism (’first return map’) ${d_i}$. If rank $({H_1}({M_1};{\mathbf {Z}})) = 1$, then $({M_1},{\phi _1})$ is ${C^s}$ equivalent $(s \leqslant r)$ to $({M_2},{\phi _2})$ if and only if ${d_1}$ is ${C^s}$ conjugate to ${d_2}$. If rank $({H_1}({M_1};{\mathbf {Z}})) \ne 1$ and ${\phi _1}$ has a periodic orbit, then there are infinitely many global cross-sections ${T_i} \subseteq {M_1}$ of ${\phi _1}$, such that the associated diffeomorphisms are pairwise nonconjugate.