Entropy and ergodic probability for differentiable dynamical systems and their bundle extensions

We answer a problem of Liao [S.T. Liao, Standard systems of differential equations and obstruction sets—from linearity to perturbations, in: System Researches, Proceedings Dedicated to the 85th Anniversary of Qian Xue-Sen, Zhejiang Education Press, Hangzhou, China, 1996, pp. 279–290 (in Chinese)]: A C 1 vector field or a C 1 diffeomorphism on an n-dimensional manifold has equal entropy with that of its bundle extensions. We also prove that each ergodic probability with simple Lyapunov spectrum has at most 2 n n ! covering probabilities on each bundle extension.