On noncontinuous chaotic functions

Abstract:

A function $f:[0,1] \to [0,1]$ is constructed such that for each two distinct points $x$ and $y$ in $[0,1]$ the sequence $\left \{ {\left | {{f^n}(x) - {f^n}(y)} \right |} \right \}_{n = 0}^\infty$ is dense in $[0,1]$. Here ${f^n}$ is the $n$th iterate of $f$. Moreover a Baire 2 function can be constructed so that the above condition is valid for all distinct $x$ and $y$ in a dense open subset of $[0,1]$.