Anantharaman–Nonnenmacher entropy conjecture for Anosov flows

Let $(M,g)$ be a Riemannian manifold, let $\varphi^t$ be its geodesic flow on $S^*M$, let $\mathcal M(\Delta_g)$ be the set of semiclassical measures of Laplace eigenfunctions, let $h_{KS}(\mu)$ be the Kolmogorov–Sinai entropy of $\mu$, and let $J^u$ denote the unstable Jacobian. Anantharaman–Nonnenmacher entropy conjecture. If $\varphi^t$ has the Anosov property, then for every $\mu\in\mathcal M(\Delta_g)$,

$$h_{KS}(\mu)\geq -\frac{1}{2}\int_{S^*M}\log J^u\,d\mu.$$

The conjecture gives a quantitative lower bound on the complexity of semiclassical measures and prevents complete concentration on dynamically simple invariant sets. In higher dimensions it remains open; the source mentions explicit bounds for locally symmetric spaces and symplectic linear maps of multidimensional tori.