Points generating the principal measure of chaos
Let \(I\) be a real compact interval and \(|I|\) its length. Given two sequences \(a_n\), \(b_n\) in \(I\) and \(t\in\mathbb{R}\), consider the sequence \({1\over n}\# \{i<n:|a_i-b_i |<t\}\) and denote its \(\liminf_{n \to\infty}\) resp. \(\limsup_{n\to\infty}\) by \(\xi_{a_n,b_n}(t)\) resp. \(\xi^*_{ a_n, b_n} (t)\). These two functions of \(t\) are nondecreasing and for any \(t\in \mathbb{R}\), \(0\leq\xi_{a_n,b_n} (t)\leq\xi^*_{a_n,b_n} (t)\leq 1\). So, \[ \mu(a_n,b_n)= {1\over |I|} \int^{|I|}_0 \bigl(\xi^*_{a_n, b_n}(t)-\xi_{a_n, b_n}(t) \bigr) dt \] is a number in \([0,1]\). For a continuous map \(f:I\to I\) put \(\mu(f,x,y)= \mu(f^nx,f^ny)\). Then \(\mu(f)= \sup_{x,y\in I}\mu(f,x,y)\) is called the principal measure of chaos. (The map \(f\) is distributionally chaotic if \(\mu(f)>0\), see \textit{B. Schweizer} and \textit{J. Smítal} [Trans. Am. Math. Soc. 344, 737-754 (1994; Zbl 0812.58062)].). The main result of the paper is that for any continuous map \(f:I\to I\) there are points \(x_0,y_0\in I\) such that \(\mu(f)=\mu (f,x_0,y_0)\). For two-dimensional maps this is not true, see \textit{M. Babilonová} [Ann. Math. Sil. 13, 33-38 (1999; Zbl 0944.37011)].
scientific article; zbMATH DE number 1848852