Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems (Q2873997)

The aim of this paper is to provide positive answers to two questions posed by Arnold about the three-dimensional helicity: {\parindent= 0.5 cm \begin{itemize} \item[i)] Is the helicity \(\mathcal{H}\) invariant under conjugation by a volume-preserving homeomorphism? More precisely, if \(X\) and \(Y\) are (exact) divergence-free vector fields, \(\varphi \) is a homeomorphism that preserves the measure induced by the volume form \(\mu \) and \(\{\varphi \circ \varphi _X^t\circ \varphi ^{-1}\}=\{\varphi _Y^t\}\), does the identity \(\mathcal{H}(X)=\mathcal{H}(Y)\) hold?NEWLINE\item [ii)] If \(\{\varphi _t\}_{0\leq t\leq 1}\) is an isotopy of volume-preserving homeomorphisms, can one define a number \(\mathcal{H}(\{\varphi _t\})\) that extends the definition for smooth isotopies?NEWLINENEWLINE\end{itemize}} A last section is devoted to higher-dimensional helicities.