LAUSR

Let M be a compact n-dimensional manifold with $$ bB_{1} $$bB1 the set of Baire-1 self-maps of M. For $$f\in bB_{1}$$f∈bB1, let $$\Omega (f)=\{\omega (x,f):x\in M\}$$Ω(f)={ω(x,f):x∈M} be the collection of $$\omega $$ω-limit sets generated by f, and $$\Lambda (f)=\cup _{x\in M}\omega (x,f)$$Λ(f)=∪x∈Mω(x,f) be the set of $$\omega $$ω-limit points of f. For a typical $$f \in bB_1$$f∈bB1, we show the following: for any $$x\in M$$x∈M, the $$\omega $$ω-limit set $$\omega (x,f)$$ω(x,f) is contained in the set of points at which f is continuous, and $$\omega (x,f)$$ω(x,f) is an $$ \infty $$∞-adic adding machine; for any $$\varepsilon >0$$ε>0, there exists a natural number K such that $$f^{k}(M)\subset B_{\varepsilon }(\Lambda (f))$$fk(M)⊂Bε(Λ(f)) whenever $$k>K$$k>K. Moreover, $$f:\Lambda (f)\rightarrow \Lambda (f)$$f:Λ(f)→Λ(f) is a bijection, and $$\Lambda (f)$$Λ(f) is closed. The Hausdorff dimension of $$\Lambda (f)$$Λ(f) is zero, and the collection of $$\omega $$ω-limit sets $$\Omega (f)$$Ω(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 $$f^{m}$$fm is an element of $$bB_{1}$$bB1 for all natural numbers m.