Abstract:The study on shift spaces in ergodic theory has been beyond the classical setting, but there is a room to discuss an extension of the Kolmogorov-Sinai entropy in the ergodic theoretical point of view.
On the other hand, the rate distortion dimension recently attracted attention in mean dimension theory because it behaves like the Kolmogorov-Sinai entropy on dynamical systems in the ``large" spaces in which the usual entropies is in general infinite.
According to these background, we investigate the connection between the Gibbs measure on the product spaces and the variational principle based on the rate distortion dimension: we concretely calculate the rate distortion dimension of the Gibbs measure on the concrete setting and it satisfies the simplest case of thermodynamical formalism based on the rate distortion dimension: the extension of the maximal measure of topological entropy.
Remark that the result shows a new phenomenon which does not hold in the classical setting.
We also discuss another variational principle under more concrete settings.
From: Mao Shinoda [view email]
[v1]
Tue, 2 Jun 2026 17:30:16 UTC (22 KB)
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