A local variational principle for random bundle transformations

We introduce local topological entropy $h_{\text{top}}(T, \mathcal{U})$ and two kinds of local measure-theoretic entropy $h_μ^{(r)-}(T,\mathcal{U})$ and $h_μ^{(r)+}(T,\mathcal{U})$ for random bundle transformations. We derive a variational inequality of random local entropy for $h_μ^{(r)+}(T,\mathcal{U})$. As an application of such relation we prove a local variational principle in random dynamical system.

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