If $f$ is Anosov and $g \sim f$, does $h(g) \geq h(f) ?$
What is the definition of g~f ?
g~f means isotopic or homotopic?
True for $f$ transitive Anosov and $g$ smooth. Indeed, passing to a cover, one can assume that the bundles of $f$ are orientable and therefore the entropy of $f$ equals the log of the spectral radius homology (Ruelle-Sullivan), since $g$ is smooth if follows that $h(g) \geq h(f)$ (Yomdim).