Dynamics and abstract computability: Computing 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.

Non-deterministic computation and the Jayne-Rogers Theorem

“…A metatheorem ( [4], [2]) then allows to remove the non-deter-minism and to obtain a deterministic algorithm. Implicitly, such an approach is exhibited in [24] by Rettinger for computability of Jordan curves, and by Galatolo, Hoyrup and Robas in [8] showing computability results for invariant measures. The present paper constitutes another application of this type.…”

Rigorous computation of invariant measures and fractal dimension for maps with contracting fibers: 2D Lorenz-like maps

“…About the general problem of computing invariant measures, it is worth to remark that some negative result are known. In [9] it is shown that there are examples of computable 2 systems without any computable invariant measure. This shows some subtlety in the general problem of computing the invariant measure up to a given error.…”

“…Computable, here means that the dynamics can be approximated at any accuracy by an algorithm, see e.g [9]. for precise definition.…”

On the computability of rotation sets and their entropies

“…The computability of Julia sets has been particularly popular, see, e.g., [18,19,9,3,4,12,11,13,10]. There are several results about the computability of certain specific measures, see [3,20] and the references therein, such as a maximal entropy measure or physical measure, the numerical computation of entropy and dimension for hyperbolic systems, see, e.g., [35] and [36] and the references therein, as well as with the computation of the topological entropy/pressure for one and multi-dimensional shift maps, see, e.g., [29,47,48,28,56,57]. To the best of our knowledge, our attempt is the first to establish computability of an entire entropy spectrum within the space of all invariant measures.…”