Hayashi’s connecting lemma

Let $f:M\to M$ be a $C^1$ diffeomorphism of the compact smooth manifold $M$, and let $p,q\in M$ be such that there exists a nonperiodic point in $\omega (p,f)\cap \alpha (q,f)$ (the intersection of the alpha limit set of $q$ with the omega limit set of $p$). Then there exists a diffeomorphism $g$, arbitrarily close to $f$ in the ${𝒞}^1$ topology of ${\mathrm{Diff}}^1(M)$, such that $q$ is in the forward orbit of $p$ through $g$, i.e. such that $g^n(p)=q$ for some $n>0$.