Hausdorff dimension of chaotic sets of interval self-maps (Q674700)

A subset \(S\) of the interval \(I\) is called a chaotic set for a continuous selfmapping \(f\) of \(I\) if \(S\) has at least two elements, and if, for any two distinct points \(x\) and \(y\) in \(S\), \(\limsup _{n\rightarrow\infty} |f^n(x)-f^n(y)|>0\) and \(\liminf _{n\rightarrow\infty}|f^n(x)-f^n(y)|=0\). It is known that \(S\) can have positive Lebesgue measure [\textit{J. Smítal}, Proc. Am. Math. Soc. 87, 54-56, (1983; Zbl 0555.26003)], or even full Lebesgue measure [\textit{M. Misiurewicz}, Lect. Notes Math. 1163, 125-130 (1985; Zbl 0625.58007)]), but generically, these sets have zero Lebesgue measure [\textit{I. Mizera}, Bull. Aust. Math. Soc. 37, 89-92 (1988; Zbl 0636.58025)]. The author proves that generically these sets are of the first Baire category and have zero Hausdorff dimension.