Einstein manifolds of non-negative sectional curvature and entropy

We show that if (M-n, g) is a closed Einstein manifold of non-negative curv ature then - log R less than or equal to pi root n-1(n-2)/2 where R is the radius of convergence of the series Sigma(i greater than or equal to 2) dim (pi(i)(M)circle times Q)t(i). If we suppose in addition that M is formal t hen we show that: dim H*(M, Q) less than or equal to [1 + exp (pi root n-1(n-2)/2)](n). These results are achieved by combining the classical Morse theory of the l oop space with a new upper bound for the topological entropy of the geodesi c flow of g in terms of the curvature tensor.