Shadowing, asymptotic shadowing and s-limit shadowing
Volume 244 / 2019
Fundamenta Mathematicae 244 (2019), 287-312
MSC: Primary 37E05; Secondary 37C50, 37B10, 54H20, 26A18.
DOI: 10.4064/fm492-5-2018
Published online: 21 December 2018
Abstract
We study three notions of shadowing: classical shadowing, limit (or asymptotic) shadowing, and s-limit shadowing. We show that classical and s-limit shadowing coincide for tent maps and, more generally, for piecewise linear interval maps with constant slopes, and are further equivalent to the linking property introduced by Chen in 1991.
We also construct a system which exhibits shadowing but not limit shadowing, and we study how shadowing properties transfer to maximal transitive subsystems and inverse limits (sometimes called natural extensions).
Where practicable, we show that our results are best possible by means of examples.
