Dynamics and abstract computability: computing invariant measures

TL;DR

This paper investigates the computability of invariant measures in dynamical systems within a general framework, establishing conditions for computability and illustrating limitations through examples of non-computable measures.

Contribution

It provides a general framework for understanding when invariant measures are computable, including conditions for the transfer operator and examples of non-computable cases.

Findings

Invariant measures as fixed points of transfer operators.

Conditions under which invariant measures are computable.

Existence of computable dynamics with non-computable invariant measures.

Abstract

We consider the question of computing invariant measures from an abstract point of view. We work in a general framework (computable metric spaces, computable measures and functions) where this problem can be posed precisely. We consider invariant measures as fixed points of the transfer operator and give general conditions under which the transfer operator is (sufficiently) computable. In this case, a general result ensures the computability of isolated fixed points and hence invariant measures (in given classes of "regular" measures). This implies the computability of many SRB measures. On the other hand, not all computable dynamical systems have a computable invariant measure. We exhibit two interesting examples of computable dynamics, one having an SRB measure which is not computable and another having no computable invariant measure at all, showing some subtlety in this kind of problems.