Syndetic proximality and scrambled sets (Q378478)

The authors present results about \textit{syndetically proximal pairs} and related scrambled sets. The syndetic proximal relation is a variant of the well-known \textit{proximal relation} and was earlier studied by \textit{J. P. Clay} [Trans. Am. Math. Soc. 108, 88--96 (1963; Zbl 0115.40301)]. The paper contains abstract theorems providing sufficient conditions for the existence of syndetically proximal pairs and a systematic study of syndetic proximal relation for various classes of dynamical systems follows. The authors consider mainly interval maps and families of symbolic systems. Recall that a pair of points \((x,y)\) in a compact metric space \((X,\rho)\) is \textit{proximal} for a continuous map \(T: X\to X\) if NEWLINE\[NEWLINE \liminf_{n\to\infty}\rho(T^n(x),T^n(y))=0. NEWLINE\]NEWLINE A pair \((x,y)\in X\times X\) is syndetically proximal if for each \(\varepsilon>0\) the set NEWLINE\[NEWLINE \{n\geq 0:\rho(T^n(x),T^n(y))<\varepsilon\} NEWLINE\]NEWLINE is syndetic, that is, has bounded gaps.