Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit

Let $M$ be a compact $n$-dimensional Riemanian manifold, End($M$) the set of the endomorphisms of $M$ with the usual $\mathcal{C}^0$ topology and $ϕ: M\to\mathbb{R}$ continuous. We prove that there exists a dense subset of $\mathcal{A}$ of End($M$) such that, if $f\in\mathcal{A}$, there exists a $f$ invariant measure $μ_{\max}$ supported on a periodic orbit that maximizes the integral of $ϕ$ among all $f$ invariant Borel probability measures.

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