The Dynamics of a Typical Measurable Function are Determined on a Zero Measure Set

Abstract

Let \(A\) be a zero measure dense \(G_{\delta }\) subset of \(I=[0,1]\), with \(% \mathcal{M}\) the set of measurable self-maps of \(I\). There exists a residual set \(\mathcal{R}\subset \mathcal{M}\) such that for each \(f\) in \(\mathcal{R}\), the range of \(f\) is contained in \(A\), and the function \(f\) is one-to-one. Moreover, there exists \(h:I\rightarrow I\), a Baire-2 function, such that \(f(x)=h(x)\) \(a.e.\), and for any \(x\in I\), the trajectory \(\tau (x,h)\) is \(\infty \)-adic, so that the \(\omega \)-limit set \(\omega (x,h)\) is a Cantor set. Since the range of \(f\) is contained in \(A\), it follows that for any \(x\) in \(I\), there exists \(y\) in \(A\) such that the trajectory \(\tau (f(x),f)=\tau (y,f)\subset A\). Speaking loosely, the dynamical structures of \(f\) are completely determined by its behavior on the set \(A\).