An integral formula for topological entropy of C∞ maps

In this paper we consider a smooth dynamical system f and give estimates of the growth rates of vector fields and differential forms in the L-p norm under the action of the dynamical system in terms of entropy, topological pressure and Lyapunov exponents. We prove a formula for the topological entropy [GRAPHICS] where Df(x)(n boolean AND) is a mapping between full exterior algebras of the tangent spaces. An analogous formula is given for the topological pressure.