Invisible Parts of Attractors

TL;DR

This paper introduces epsilon-invisible sets in dynamical systems, showing that for generic systems, large parts of attractors are practically never observed due to extremely small visitation frequencies.

Contribution

The paper constructs an open set of dynamical systems with epsilon-invisible parts of attractors, demonstrating their existence in generic C^1 perturbations of a skew product over the Smale-Williams solenoid.

Findings

Large portions of attractors are almost never visited by orbits.

Epsilon-invisible sets can be of size comparable to the entire attractor.

Such sets exist in an open set of C^1 perturbations of a specific skew product.

Abstract

This paper deals with the attractors of generic dynamical systems. We introduce the notion of epsilon-invisible set, which is an open set in which almost all orbits spend on average a fraction of time no greater than epsilon. For extraordinarily small values of epsilon (say, smaller than 2^{-100}), these are areas of the phase space which an observer virtually never sees when following a generic orbit. We construct an open set in the space of all dynamical systems which have an epsilon-invisible set that includes parts of attractors of size comparable to the entire attractor of the system, for extraordinarily small values of epsilon. The open set consists of C^1 perturbations of a particular skew product over the Smale-Williams solenoid. Thus for all such perturbations, a sizable portion of the attractor is almost never visited by generic orbits and practically never seen by the observer.