Summary: Let \(X\) be a compact connected metric space and \(f : X \rightarrow X\) be a continuous map. In this paper, we prove that if \(f\) has a periodic point and \(\omega_f\) is continuous then the almost periodic set is a finite union of cyclically permuted subcontinua of \(X\). In particular, \(AP(f)\) is connected whenever \(f\) has a fixed point. Also we show that for dendrites with closed endpoint set, if \(\omega_f(a)\) is infinite, then \(\omega_f\) is continuous at \(a\) if, and only if, \(f\) is equicontinuous at \(a\). We show that the later result fails whenever \(\omega_f(a)\) is finite or the endpoints set is not closed. We give an example of a local dendrite map \(f : X \rightarrow X\) for which \(\omega_f\) is continuous, \(f_{|X_{\infty}}\) is equicontinuous but \(f\) is not equicontinuous on the whole space \(X\). Finally, we answer to an open question raised by G. Acosta and D. Fernández-Bretón [Fundam. Math. 254, No. 2, 215–240 (2021; Zbl 1514.37056)] by providing a class of dendrites on which the equicontinuity of \(f_{|X_{\infty}}\) imply the equicontinuity of \(f\).