Let $f:M\to M$ be a $C^1$ diffeomorphism of a compact Riemannian manifold, and let $\mu$ be an $f$-invariant probability measure. Let $\lambda_i(x)$ be the Lyapunov exponents of the derivative cocycle $df_x:T_xM\to T_{f(x)}M$ given by Oseledets' theorem, with multiplicities $m_i(x)$. Then the measure-theoretic entropy satisfies

\begin{align*} h_\mu(f)\leq \int_M \sum_{\lambda_i(x)>0} m_i(x)\lambda_i(x)\,d\mu(x). \end{align*}

For ergodic $\mu$, this becomes

\begin{align*} h_\mu(f)\leq \sum_{\lambda_i>0} m_i\lambda_i. \end{align*}