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Realizing semicomputable simplices by computable dynamical systems

Abstract : We study the computability of the set of invariant measures of a computable dynamical system. It is known to be semicomputable but not computable in general, and we investigate which semicomputable simplices can be realized in this way. We prove that every semicomputable finite-dimensional simplex can be realized, and that every semicomputable finite-dimensional convex set is the projection of the set of invariant measures of a computable dynamical system. In particular, there exists a computable system having exactly two ergodic measures, none of which is computable. Moreover, all the dynamical systems that we build are minimal Cantor systems.
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https://hal.inria.fr/hal-03339422
Contributor : Mathieu Hoyrup Connect in order to contact the contributor
Submitted on : Thursday, September 9, 2021 - 2:20:00 PM
Last modification on : Thursday, February 3, 2022 - 11:16:32 AM
Long-term archiving on: : Saturday, December 11, 2021 - 7:23:33 AM

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  • HAL Id : hal-03339422, version 1

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Daniel Coronel, Alexander Frank, Mathieu Hoyrup, Cristóbal Rojas. Realizing semicomputable simplices by computable dynamical systems. 2021. ⟨hal-03339422⟩

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