A chaotic function with a distributively scrambled set of full Lebesgue measure (Q876936)

For a map \(f:[0,1]\to[0,1]\), a subset \(J\subset[0,1]\) is said to be scrambled if every \(x\in J\) has the property that \(\limsup_{n\to\infty}| f^n(x)-f^n(y)|>0\) for any \(x\neq y\in J\) or \(y\in[0,1]\) periodic under \(f\) and at the same time \(\liminf_{n\to\infty}| f^n(x)-f^n(y)|=0\) for all \(y\in[0,1]\). \textit{M. Misiurewicz} [in: Iteration theory and its functional equations. Lect. Notes Math. 1163, 125--130 (1985; Zbl 0625.58007)] exhibited a continuous map with a scrambled set of full Lebesgue measure. A (strictly) stronger definition -- `distributively' or `strongly' scrambled -- appeared in the work of \textit{B. Schweizer} and \textit{J. Smítal} [Trans. Am. Math. Soc. 344, No. 2, 737--754 (1994; Zbl 0812.58062)]. In this paper a continuous map is constructed which has a distributively scrambled set in which each point has dense orbit.