Positive topological entropy implies chaos DC2 (Q2862178)

Li and Yorke invented the idea of chaos in 1975. This idea was strengthened by \textit{B. Schweizer} and \textit{J. Smítal} in 1994 [Trans. Am. Math. Soc. 344, No. 2, 737--754 (1994; Zbl 0812.58062)] and later it evolved into three definitions of distributional chaos: DC1, DC2, and DC3. This paper tackles the relationship between a well-known invariant, namely topological entropy, and DC2. The author proves the following theorem:NEWLINENEWLINE{Main Theorem: } Suppose that \((X,T)\) is a topological dynamical system with positive topological entropy. Then, there exists an uncountable DC2-scrambled set \(E\subset X\).NEWLINENEWLINEThe author uses an equivalent definition of DC2-scrambled that is expressed in terms of ergodic averages.NEWLINENEWLINE{Definition:} A pair \(x,y\in X\) is DC2-scrambled ifNEWLINENEWLINE(1) \(\liminf_{n\rightarrow\infty} \frac{1}{n}\sum_{i=1}^n d(T^ix,T^iy) =0\)NEWLINENEWLINEandNEWLINENEWLINE(2) \(\limsup_{n\rightarrow\infty} \frac{1}{n}\sum_{i=1}^n d(T^ix,T^iy) >0\).NEWLINENEWLINEThe author then goes on to prove the topological statement (of the main theorem) using ergodic-theoretic tools.