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A fast dynamo with zero topological entropy

  • Lukas Niebel
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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 L2. For the same velocity field, every ideal solution grows at most linearly in L2. 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 C1.

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Cite

BibTeXRIS
BibTeX
@misc{niebel2026fast,
  title = {{A fast dynamo with zero topological entropy}},
  author = {Lukas Niebel},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.02586},
  eprint = {2608.02586},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.02586},
}
RIS
TY  - UNPB
TI  - A fast dynamo with zero topological entropy
AU  - Lukas Niebel
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2608.02586
UR  - https://arxiv.org/abs/2608.02586
AB  - 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 L^2. For the same velocity field, every ideal solution grows at most linearly in 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 C^1.
ER  -