Dynamics and abstract computability: Computing invariant measures
Stefano Galatolo Mathieu Hoyrup Cristóbal Rojas
Discrete & Continuous Dynamical Systems - A 2011, 29(1): 193-212 doi: 10.3934/dcds.2011.29.193
We consider the question of computing invariant measures from an abstract point of view. Here, computing a measure means finding an algorithm which can output descriptions of the measure up to any precision. We work in a general framework (computable metric spaces) where this problem can be posed precisely. We will find invariant measures as fixed points of the transfer operator. In this case, a general result ensures the computability of isolated fixed points of a computable map.
     We give general conditions under which the transfer operator is computable on a suitable set. This implies the computability of many "regular enough" invariant measures and among them many physical measures.
     On the other hand, not all computable dynamical systems have a computable invariant measure. We exhibit two examples of computable dynamics, one having a physical measure which is not computable and one for which no invariant measure is computable, showing some subtlety in this kind of problems.
keywords: Physical Measure Computable Analysis Computation Dynamical System.

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