Let \(f\) be an expansive homeomorphism of a compact metric space \(X\), and assume that \(f\) has the specification property. A real-valued function \(\phi\) on \(X\) with appropriate properties will uniquely determine an equilibrium state (Gibbs measure) \(\mu_\phi\), and one can discuss the local entropy function on \(X\) that is determined by \(\mu_\phi\). The central objects studied in the paper are the topological entropies of the restrictions of \(f\) to the level sets of the local entropy function.
The main result is a variational relation between this multifractal spectrum and other thermodynamical properties. The results are similar in form to results contained in [L. Barreira, Ya. Pesin and J. Schmeling, Chaos 7, No. 1, 27-38 (1997; Zbl 0933.37002)], which extend earlier results by Ya. Pesin and H. Weiss. The focus of this earlier work was on local dimensions rather than local entropy.