Abstract: In a joint work with Martin Sambarino, we prove that if $S$ is a smooth compact boundaryless

orientable surface of genus $g$, furnished with a smooth area form $\omega$ and $\Diff^r_{\omega}(S)$,

$1\leq r\leq +\infty$, is the space of $C^r$ diffeomorphisms of $S$, then generically in  $\Diff^r_{\omega}

(S)$, every hyperbolic periodic point has a transverse homoclinic intersection. Consequently the topological

entropy is positive on an open and dense set of $\Diff^r_{\omega}(S)$. The proof is divided into three steps. 

i) proving that under ``explicit classical generic properties'' the conclusion occurs if there are more than $2g-2$ periodic points;

ii) proving that if these generic properties are satisfied and there are exactly $2g-2$ periodic points, then there is a power of $f$ that is isotopic to the identity (at least if $g\geq 2);

iii) proving that if $f\in\Diff^r_{\omega}(S)$ is isotopic to the identity, it can be perturbed to $g\in\Diff^r_{\omega}(S)$ with more than $2g-2$ periodic points.