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 C1 vector field or a C1 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 n2n! covering probabilities on each bundle extension.