Amorphic complexity

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.

Constant length substitutions, iterated function systems and amorphic complexity

“…Theorem 3.1 ( [FGJ16]). If a dynamical system (X, f ) has positive topological entropy or is weakly mixing with respect to some invariant probability measure µ with non-trivial support, then (X, f ) has infinite separation numbers.…”

“…We refer the reader to [FGJ16] for an in-depth discussion of amorphic complexity and several classes of examples. However, the following statement is worth recalling.…”

“…In this article we provide this kind of relation for the topological notion of amorphic complexity which was recently introduced in [FGJ16] to study dynamical systems with zero entropy. More precisely, we show that amorphic complexity of symbolic systems coincides with the box dimension of an associated subset in the so-called Besicovitch space of (A Z , σ) endowed with a canonical metric, see Lemma 6.2 and Theorem 6.5.…”

Some remarks on modified power entropy

“…Otherwise, it is again possible to define an upper and a lower amorphic complexity. Basic properties of this quantity, like topological invariance, factor relations, power invariance and a product rule, as well as the application to a number of example classes are discussed in [FGJ15]. Somewhat surprisingly, amorphic complexity turns out be very well applicable and accessible to explicit computations in various system classes like regular Toeplitz flows, Sturmian shifts and Denjoy type circle homeomorphisms or cut and project quasicrystals.…”

“…In this context, we also introduce and discuss amorphic complexity. This is a new topological invariant that equally measures the complexity of zero entropy systems, but is based on an asymptotic rather than a finite-time concept of separation [FGJ15].…”