Real Analysis Exchange

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

T. H. Steele

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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 \(A\).

Article information

Source
Real Anal. Exchange, Volume 45, Number 2 (2020), 375-386.

Dates
First available in Project Euclid: 30 June 2020

Permanent link to this document
https://projecteuclid.org/euclid.rae/1593482446

Digital Object Identifier
doi:10.14321/realanalexch.45.2.0375

Subjects
Primary: 26A18: Iteration [See also 37Bxx, 37Cxx, 37Exx, 39B12, 47H10, 54H25]
Secondary: 54H20: Topological dynamics [See also 28Dxx, 37Bxx] 37B20: Notions of recurrence

Keywords
measurable function ω-limit set trajectory generic

Citation

Steele, T. H. The Dynamics of a Typical Measurable Function are Determined on a Zero Measure Set. Real Anal. Exchange 45 (2020), no. 2, 375--386. doi:10.14321/realanalexch.45.2.0375. https://projecteuclid.org/euclid.rae/1593482446


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