A fast dynamo with zero topological entropy

Abstract

We construct an autonomous Lipschitz fast dynamo on the flat three-torus whose particle flow has zero topological entropy and whose ideal induction equation has no exponential growth. For every sufficiently small positive magnetic diffusivity, the induction operator has an eigenvalue with real part bounded below by a positive constant independent of the diffusivity. A corresponding non-zero real-valued magnetic field satisfies an exact exponential growth law in ${\mathrm{L}}^2$. For the same velocity field, every ideal solution grows at most linearly in ${\mathrm{L}}^2$. Moreover, the particle flow and its inverse have Lipschitz constants growing at most linearly in time. The velocity field is real-valued, divergence-free, differentiable everywhere and smooth away from one circle, but is not ${\mathrm{C}}^1$.

arXiv commentTitle changed; proof substantially simplified; results unchanged. Added a reference to arXiv:2609.04153, which resolves Arnold's fast dynamo conjecture