For a C-infinity diffeomorphism of a closed surface, a weak-star limit of ergodic measures splits as mu = (1-beta)mu_0 + beta*mu_1, where the neutral part mu_0 is carried by long orbit segments along which the unstable direction fails to expand. The neutral part contributes nothing to the top Lyapunov exponent, so the limit of the exponents equals beta times the exponent of the positive part; reparametrization estimates then show those same neutral segments create no entropy either, giving the ratio inequality lim h(nu_k)/h(mu) <= lim lambda+(nu_k)/lambda+(mu) <= 1 that ties the two continuity defects together. Consequences a student can follow include upper semicontinuity of Hausdorff dimension over ergodic measures with entropy bounded away from zero, a characterization of when a hyperbolic SRB measure exists, and continuity of the top exponent along measures whose entropy tends to the topological entropy.
This paper by Jerome Buzzi examines the relationship between entropy and Lyapunov exponents - two fundamental numerical invariants in measure-theoretic ergodic theory. The work investigates how entro…