Entropy versus volume via Heegaard diagrams

The following inequalities are established, improving a former inequality due to Kojima. For any closed arithmetic hyperbolic $3$--manifold fibering over a circle, the entropy of the pseudo-Anosov monodromy is bounded by the hyperbolic volume of the $3$--manifold, up to a universal constant factor. For any closed hyperbolic $3$--manifold fibering over a circle with systole $\geq\varepsilon>0$, the entropy is bounded by the hyperbolic volume times $\log(3+1/\varepsilon)$, up to a universal constant factor. The proof relies on Heegaard Floer homology and hyperbolic geometry.