Abstract
We obtain a lower bound for the entropy of a Borel probability measure (not necessarily invariant) with respect to an upper semicontinuous set-valued map as the product of the lower dimension of the measure and the logarithmic expansion rate. This is a generalization of the well-known measure-preserving single-valued case.
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 178, Optimal Control, 2020.
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Carrasco-Olivera, D., Metzger, R. & Morales, C.A. Logarithmic Expansion, Entropy, and Dimension for Set-Valued Maps. J Math Sci 276, 227–236 (2023). https://doi.org/10.1007/s10958-023-06737-y
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DOI: https://doi.org/10.1007/s10958-023-06737-y