Amorphic complexity can take any nonnegative value in general metric spaces. Issue 1 (2nd January 2021)

• Record Type: Journal Article
• Authors: Kulczycki, Marcin
• Is Part Of: Dynamical systems. Volume 36:Issue 1(2021)
• Page Start: 64
• Page End: 68
• Publication Date: 2021-01-02
• Subjects: Amorphic complexity -- entropy zero; Primary 37B99 -- secondary 37A35 -- 37C70; Differentiable dynamical systems -- Periodicals; 515.35205
• DOI: 10.1080/14689367.2020.1806990
• Languages: English
• ISSNs: 1468-9367

Abstract

Abstract : Amorphic complexity, introduced in 2016, is a new topological invariant of discrete dynamical systems that is of particular interest for systems of entropy zero. Examples exist that show that amorphic complexity can take values 0 and 1 or more (inlcuding infinity), but it was not known if values from the interval (0, 1) could be taken. This note proves that for every real nonnegative number there exists a noncomplete metric space and a map on it that has amorphic complexity equal to this number. The construction makes strong use of incompleteness of the space and the question whether these values can be realized for a map on a compact metric space remains open.