Entropy of orthonormal \(n\)-frame flows (Q2730684)

The author gives definitions for the topological and measure theoretic entropies of flows which are generalizations of the definitions of Bowen's and Katok's definitions [\textit{R. Bowen}, Trans. Am. Math. Soc. 153, 401-414 (1971; Zbl 0212.29201) and \textit{A. Katok}, Publ. Math., Inst. Hautes Étud. Sci. 51, 137-173 (1980; Zbl 0445.58015)]. He proves that these definitions are equivalent to defining the entropies of the flow in terms of its time-one map. Using the new definitions he answers a question of Liao concerning the relationship between the entropy of a flow and the entropy of its lift to the bundle of orthonormal frames [\textit{S. Liao}, Standard systems of differential equations and obstruction sets - from linearity to perturbations. Proceedings of the 1983 Beijing symposium on differential geometry and differential equations, Science Press, Beijing 65-97 (1986; Zbl 0632.00008)]. He proves that the measure theoretic entropies of the flow and its lift are always the same, and that the topological entropies are preserved in case the lifted flow is Liao hyperbolic. The author proves the analogous result for lifts of diffeomorphisms to the bundle of orthonormal frames in his article [Entropy for frame bundle systems and Grassmann bundle systems induced by a diffeomorphism, Sci. China, Ser. A 45, No. 9, 1147-1153 (2002)].