Amorphic complexity

TL;DR

Amorphic complexity is a new topological invariant that measures the onset of disorder in zero-entropy dynamical systems, distinguishing subtle differences in their asymptotic behavior.

Contribution

It introduces amorphic complexity as a novel invariant, linking it to asymptotic separation, box dimension, and scaling behavior in various dynamical systems.

Findings

Amorphic complexity detects disorder in zero-entropy systems.

It distinguishes almost automorphic from equicontinuous systems.

For symbolic systems, it equals the box dimension of the Besicovitch space.

Abstract

We introduce amorphic complexity as a new topological invariant that measures the complexity of dynamical systems in the regime of zero entropy. Its main purpose is to detect the very onset of disorder in the asymptotic behaviour. For instance, it gives positive value to Denjoy examples on the circle and Sturmian subshifts, while being zero for all isometries and Morse-Smale systems. After discussing basic properties and examples, we show that amorphic complexity and the underlying asymptotic separation numbers can be used to distinguish almost automorphic minimal systems from equicontinuous ones. For symbolic systems, amorphic complexity equals the box dimension of the associated Besicovitch space. In this context, we concentrate on regular Toeplitz flows and give a detailed description of the relation to the scaling behaviour of the densities of the p-skeletons. Finally, we take a look at strange non-chaotic attractors appearing in so-called pinched skew product systems. Continuous-time systems, more general group actions and the application to cut and project quasicrystals will be treated in subsequent work.