Volume growth and topological entropy of certain partially hyperbolic systems

Let $f$ be a $C^1$ diffeomorphism on a compact manifold $M$ admitting a partially hyperbolic splitting $TM=E^{s}\oplus_{\prec} E^{1}\oplus_{\prec} E^{2}\cdots \oplus_{\prec}E^{l}\oplus_{\prec} E^{u}$where$E^{s}$ is uniformly contracting, $E^u$ is uniformly expanding and $\dim E^i=1,1\le i\le l\mathrm{.}$ We prove an entropy formula w.r.t. the volume growth rate of subspaces in the tangent bundle: $$h_{\mathrm{t}\mathrm{o}\mathrm{p}}(f)=\underset{n\to +\mathrm{\infty }}{\lim }\frac{1}{n}\log \int \underset{V\subset T_{x}M}{\max }\mathrm{\mid }\det D{f}_x^n{\mathrm{\mid }}_V\mathrm{\mid }dx\mathrm{.}$$