Unstable Entropy and Volume Growth for Partially Hyperbolic Diffeomorphisms

Let f be a C¹ diffeomorphism on a compact manifold M admitting a partially hyperbolic splitting TM=Eᶜˢ⊕ Eᵘ where Eᵘ is uniformly expanding. We establish a lower bound for the unstable topological entropy (w.r.t. Eᵘ): hᵘₜₒₚ(f)≥n→+∞1/nlog∫M| (Dfⁿₓ|_Eᵘ)|\,dx. Moreover, if there exists a measure of maximal unstable entropy whose Lyapunov exponents along Eᶜˢ are all non-positive, then equality holds and the is in fact a limit.