For Bedford-McMullen sponge systems with r coordinates, the metric mean dimension is a weighted sum of topological entropies of projections, and the mean Hausdorff dimension equals a weighted topological entropy divided by log m_1.
Variational principles of relative weighted topological pressures
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abstract
Recently, M. Tsukamoto (New approach to weighted topological entropy and pressure, Ergod. Theory Dyn. Syst. 43 (2023) 1004-1034) used a new approach to define the weighted topological entropy and pressure. Inspired by his ideas, we introduce the relative weighted topological entropy and pressure for factor maps and establish several variational principles. One of these results involves a question raised by D. Feng and W. Huang (Variational principle for weighted topological pressure, J. Math. Pures Appl. 106 (2016) 411-452), whether there is a relative version of the weighted variation principle. In this paper, we try to establish such variational principle. Furthermore, we generalize the Ledrappier and Walters' type relative variational principle to the weighted version.
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Mean dimension theory for infinite dimensional Bedford-McMullen sponges
For Bedford-McMullen sponge systems with r coordinates, the metric mean dimension is a weighted sum of topological entropies of projections, and the mean Hausdorff dimension equals a weighted topological entropy divided by log m_1.