Volume growth and topological entropy of certain partially hyperbolic systems

Abstract

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 $\dim E^i=1,1\le i\le l.$ We prove an entropy formula w.r.t. the volume growth rate of subspaces in the tangent bundle: $$h_{\mathrm{top}}(f)=\underset{n\to +\infty }{\lim }\frac{1}{n}\log \int \underset{V\subset T_{x}M}{\max }|\det D{f}_x^n{|}_V|dx.$$