Dynamics of Baire-1 functions on the interval

Research Article

Abstract

Let with $$bB_{1}$$ the set of Baire-1 self-maps of I. For , let be the collection of $$\omega$$-limit sets generated by f, and $$\mathrm{\Lambda } (f)=\bigcup _{x\in I}\omega (x,f)$$ be the set of $$\omega$$-limit points of f. There exists S a residual subset of $$bB_{1}$$ such that for any , the following hold:
• For any $$x\in I$$, 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 M such that $$f^{m}(I)\subset B_{\varepsilon }(\mathrm{\Lambda } (f))$$ whenever $$m>M$$. Moreover, $$f:\mathrm{\Lambda } (f)\rightarrow \mathrm{\Lambda } (f)$$ is a bijection, and $$\mathrm{\Lambda } (f)$$ is closed.

• The Hausdorff s-dimensional measure of $$\mathrm{\Lambda } (f)$$ is zero for all $$s>0$$.

• The collection of $$\omega$$-limit sets $$\mathrm{\Omega } (f)$$ is closed in the Hausdorff metric space.

• If x is a point at which f is continuous, then (xf) is a point at which the map given by $$(x,f)\mapsto \omega (x,f)$$ is continuous.

• The n-fold iterate $$f^{n}$$ is an element of $$bB_{1}$$ for all natural numbers n.

• The function f is non-chaotic in the senses of Devaney and Li–Yorke.

Keywords

Baire-1 function $$\omega$$-Limit set Odometer Generic

Mathematics Subject Classification

54H20 37B99 26A18

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