Dynamics of Baire-1 functions on the interval

  • Timothy H. Steele
Research Article


Let Open image in new window with \(bB_{1}\) the set of Baire-1 self-maps of I. For Open image in new window , let Open image in new window 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 Open image in new window , 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 Open image in new window 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.


Baire-1 function \(\omega \)-Limit set Odometer Generic 

Mathematics Subject Classification

54H20 37B99 26A18 


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Authors and Affiliations

  1. 1.Department of MathematicsWeber State UniversityOgdenUSA

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