Summary: Let \(M\) be a compact \(n\)-dimensional manifold with \(bB_1\) the set of Baire-1 self-maps of \(M\). For \(f\in bB_1\), let \(\Omega (f)=\{\omega (x,f):x\in M\}\) be the collection of \(\omega \)-limit sets generated by \(f\), and \(\Lambda (f)=\cup _{x\in M}\omega (x,f)\) be the set of \(\omega \)-limit points of \(f\). For a typical \(f \in bB_1\), we show the following: for any \(x\in M\), the \(\omega \)-limit set \(\omega (x,f)\) is contained in the set of points at which \(f\) is continuous, and \(\omega (x,f)\) is an \(\infty \)-adic adding machine; for any \(\varepsilon >0\), there exists a natural number \(K\) such that \(f^k(M)\subset B_{\varepsilon }(\Lambda (f))\) whenever \(k>K\). Moreover, \(f:\Lambda (f)\rightarrow \Lambda (f)\) is a bijection, and \(\Lambda (f)\) is closed. The Hausdorff dimension of \(\Lambda (f)\) is zero, and the collection of \(\omega \)-limit sets \(\Omega (f)\) is closed in the Hausdorff metric space. The function \(f\) is not chaotic in the sense of Li-Yorke, nor in the sense of Devaney. The function \(f\) is one-to-one, and the \(m\)-fold iterate \(f^m\) is an element of \(bB_1\) for all natural numbers \(m\).