On the entropy of conservative flows

We obtain a $C^1$-generic subset of the incompressible flows in a closed three-dimensional manifold where Pesin's entropy formula holds thus establishing the continuous-time version of \cite{T}. Moreover, in any compact manifold of dimension larger or equal to three we obtain that the metric entropy function and the integrated upper Lyapunov exponent function are not continuous with respect to the $C^1$ Whitney topology. Finally, we establish the $C^2$-genericity of Pesin's entropy formula in the context of Hamiltonian four-dimensional flows.Comment: 10 page

A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion

We provide a systematic approach for deducing statistical limit laws via martingale-coboundary decomposition, for nonuniformly hyperbolic systems with slowly contracting and expanding directions. In particular, if the associated return time function is square-integrable, then we obtain the central limit theorem, the weak invariance principle, and an iterated version of the weak invariance principle.Comment: Final versio