Preprint D37/2006
On Cr-closing for flows on orientable and non-orientable 2-manifolds

Benito Pires | Gutierrez, Carlos

**Keywords: **
recurrence | Cr-closing | Cr-connecting | structural stability

We show that on any compact surface, whether orientable or not,
supporting non--trivial recurrence, there exists a large class of flows for whom $C^r-$closing by arbitrarily small
$C^r-$twist--perturbations holds. More specifically, let
$X\in\mathfrak{X}^r(M)$ be a $C^r$ vector field which has
singularities, all of which hyperbolic, and let $P$ be the holonomy map around a non--trivial recurrent point $p$ of $X$. We say that $X$ is dissipative at a saddle $s\in M$ if the trace of the derivative of $X$ at $s$ is negative, and a set $B\subset{\rm dom}(P)$ (the domain of $P$) is of total measure if $\nu(B)=1$ for any $P-$invariant Borel probability measure $\nu$. We prove the following: if $X$ is dissipative at its saddles and $\liminf_{n\to\infty}\frac1n\log\vert DP^n(x)\vert<0$ on a set of total measure then there exists $Y\in\mathfrak{X}^r(M)$ arbitrarily close to $X$ in the $C^r-$topology having a periodic trajectory passing \mbox{through $p$}.