Hyperbolicity of maximal entropy measures for certain maps isotopic to Anosov diffeomorphisms (Q6115677)

Consider a continuous transformation \(f : M \to M\), where \(M\) is a compact metric space. An ergodic invariant probability measure \(\mu\) such that \(h_{\mu}(f) = h_{\mathrm{top}}(f)\) is called a \textit{maximal entropy measure} for \(f\), where \(h_{\mu}(f)\) is the entropy with respect to \(\mu\) and \(h_{\mathrm{top}}(f)\) is the topological entropy. Such measures describe the complexity of the whole system. The author shows that for a class of partially hyperbolic diffeomorphisms on \(\mathbb{T}^d\), which have compact two-dimensional center foliations, the maximal entropy measures (if exist) are \textit{hyperbolic}, that is, they have no zero Lyapunov exponents and there exist Lyapunov exponents with different signs.