Entropy and self-intersection number of geodesic currents on compact hyperbolic surfaces

Let \$X\$ be a compact hyperbolic surface and \$C\$ a geodesic current which is a geodesic-flow invariant measure. Denote the measure-theoretic entropy of \$C\$ by \$h_X​(C)\$. In this talk, assuming \$C\$ is ergodic, we give an upper bound on \$h_X​(C)\$ in terms of its self-intersection number \$i(C,C)\$ and the systole of \$X\$. In particular, we show that a small self-intersection number forces small entropy.