How much information about a dynamical system do its recurrences contain?

TL;DR

This paper demonstrates that, given certain conditions, the recurrence patterns of a dynamical system encode enough information to uniquely determine its topological structure, establishing a link between recurrences and system topology.

Contribution

It proves that Poincare recurrences can fully determine the topology of a dynamical system under specific assumptions, highlighting the informational power of recurrence data.

Findings

Recurrences determine the system's topology.

Systems with identical recurrences are topologically equivalent.

The result holds under suitable assumptions.

Abstract

We show that, under suitable assumptions, Poincare recurrences of a dynamical system determine its topology in phase space. Therefore, dynamical systems with the same recurrences are topologically equivalent.