On the entropy of conservative flows

TL;DR

This paper proves the generic validity of Pesin's entropy formula for certain conservative flows and shows the discontinuity of entropy and Lyapunov exponents in specific topologies, with results in both three and four dimensions.

Contribution

It establishes the $C^1$-generic validity of Pesin's entropy formula for incompressible flows and the $C^2$-genericity for Hamiltonian flows, advancing understanding of entropy in conservative dynamical systems.

Findings

Pesin's entropy formula holds for a $C^1$-generic subset of incompressible flows in 3D.

The metric entropy and Lyapunov exponent functions are discontinuous in the $C^1$ Whitney topology in higher dimensions.

Pesin's entropy formula is $C^2$-generically valid for Hamiltonian flows in 4D.

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 \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.