## Real Analysis Exchange

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

T. H. Steele

#### 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