Conjugacies of Expanding Skew Products on Tⁿ

We show that any equilibrium state for a H\"older potential on the model map x⃗ ↦ d · x⃗ Zⁿ on Tⁿ is conjugate to Lebesgue measure for an invariant expanding skew product of degree d. This is a generalization of a result of McMullen to higher dimensions for equilibrium states. We use an approach developed by the author using a family of nonstationary transfer operators for an expanding skew product. We also apply a Markov partition argument to classify invariant probability measures for expanding maps on Tⁿ.