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 $\ge \epsilon >0$, the entropy is bounded by the hyperbolic volume times $\log (3+1\mathrm{/}\epsilon )$, up to a universal constant factor. The proof relies on Heegaard Floer homology and hyperbolic geometry.