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Topological and metric entropies are among the most important global invariants of smooth dynamical systems. Topological entropy characterizes the total exponential complexity of the orbit structure with a single number. Metric entropy with respect to an invariant measure gives the exponential growth rate of the statistically significant orbits. The knowledge of entropies, especially in low-dimensional cases, provides a wealth of quantitative structural information about the system. Such information includes the growth rate of the number of periodic orbits [K 1], "large horseshoes" [K 3], the growth rate of the volume of cells of various dimensions ([Mn] for geodesic flows, [Y] in general), ergodic components, and factors with very stochastic behavior [Pe, Si], etc. Unfortunately, because the entropies are defined in global asymptotic terms, a priori, we do not expect them to change smoothly when the system is perturbed, even in a very nice topology like C°° or C (real analytic). There are several results concerning the regularity of entropy for general diffeomorphisms and flows on compact manifolds. Misiurewicz [Mi 1] constructed examples to show that the topological entropy, h ' Diffoo(M") —• R is not continuous for n > 4 . It seems unknown, although unlikely, whether entropy is continuous for n = 3 . Yomdin [Y] and Newhouse [N] have proved that ht : DiSQO(M ) —> R is upper-semicontinuous for n>2. Katok

Differentiability of entropy for Anosov and geodesic flows