On the Entropy of Conservative Flows

Abstract

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 Tahzibi (C R Acad Sci Paris I 335: [1057][1058][1059][1060][1061][1062] 2002). 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.

Lyapunov exponents and entropy for divergence-free Lipschitz vector fields

“…Therefore, we are left with possible generalizations considering other classes of conservative dynamical systems, e.g., symplectic homeomorphisms, Hamiltonians, contact flows, geodesic flows, etc. In this direction [10] proves that C 2 -generic Hamiltonians, i.e. C 1 -generic Hamiltonian flows, satisfy the Pesin entropy formula.…”

“…We denote by h ν (X t ) the measure-theoretic entropy of the time-t map X t w.r.t. the measure ν (for more details see [10]). Since Abramov [1] formula says that h ν (X t ) = |t| h ν (X 1 ) for any t ∈ R (for a proof see [13, Theorem 3, p. 255]), it is irrelevant to choose other time-t flow to evaluate the metric entropy.…”

“…Given a measure ν and a ν-measurable flow X t , we denote by h ν (X 1 ) the measure-theoretic entropy of the time-1 map X 1 with respect to (w.r.t.) the measure ν (for more details see [10]). It is worth pointing out that several authors defined different concepts of flow entropy which are well behaved when we consider a re-parametrization of the flow [14,37].…”