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On the Closing Lemma problem for the torus

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  • We investigate the open Closing Lemma problem for vector fields on the $2$-dimensional torus. The local $C^r$ Closing Lemma is verified for smooth vector fields that are area-preserving at all saddle points. Namely, given such a $C^r$ vector field $X$, $r\geq 4$, with a non-trivially recurrent point $p$, there exists a vector field $Y$ arbitrarily near to $X$ in the $C^r$ topology and obtained from $X$ by a twist perturbation, such that $p$ is a periodic point of $Y$.
       The proof relies on a new result in $1$-dimensional dynamics on the non-existence of semi-wandering intervals of smooth order-preserving circle maps.
    Mathematics Subject Classification: Primary: 34D30; Secondary: 37E35, 37C10.


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