Syndetic proximality and scrambled sets. (English) Zbl 1327.37006

Moothathu, T. K. Subrahmonia; Oprocha, Piotr

Topol. Methods Nonlinear Anal. 41, No. 2, 421-461 (2013).

The authors present results about syndetically proximal pairs and related scrambled sets. The syndetic proximal relation is a variant of the well-known proximal relation and was earlier studied by 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 proximal for a continuous map \(T: X\to X\) if \[ \liminf_{n\to\infty}\rho(T^n(x),T^n(y))=0. \] A pair \((x,y)\in X\times X\) is syndetically proximal if for each \(\varepsilon>0\) the set \[ \{n\geq 0:\rho(T^n(x),T^n(y))<\varepsilon\} \] is syndetic, that is, has bounded gaps.

Reviewer: Dominik Kwietniak (Krakow)

MSC:

37B05	Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B10	Symbolic dynamics

Keywords:

syndetically proximal pair; proximal relation; scrambled set; Li-Yorke chaos; entropy; transitivity; subshift; interval maps; symbolic dynamics

Citations:

Zbl 0115.40301