Entropy and periodic points in pseudo-Anosov isotopy classes

Let $f$ be a pseudo-Anosov homeomorphism of a closed orientable surface of genus at least two. We prove that, for every $C^\infty$ diffeomorphism $g$ isotopic to $f$, the exponential growth rate of the number of nonempty Nielsen classes of its iterates equals $h_{\mathrm{top}}(g)$ if and only if $h_{\mathrm{top}}(g)=h_{\mathrm{top}}(f)$. Thus Nielsen growth captures the full entropy precisely for representatives minimizing entropy in their isotopy class. For $C^{1+α}$ diffeomorphisms, $α>0$, we identify this growth rate with the entropy of the global orbit quotient in Fathi's hyperbolic extension and establish Bowen's periodic-point formula for the quotient. We also prove that quotient entropy is upper semicontinuous in the $C^0$ topology throughout the isotopy class and continuous at every $C^{1+α}$ diffeomorphism, even under perturbations by arbitrary homeomorphisms in the class.