Normal form for maps with nilpotent linear part

Fahimeh Mokhtari*, Ernst Roell, Jan A. Sanders

*Corresponding author for this work

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The normal form for an n-dimensional map with irreducible nilpotent linear part is determined using sl2-representation theory. We sketch by example how the reducible case can also be treated in an algorithmic manner. The construction (and proof) of the sl2-triple from the nilpotent linear part is more complicated than one would hope for, but once the abstract sl2 theory is in place, both the description of the normal form and the computational splitting to compute the generator of the coordinate transformation can be handled explicitly in terms of the nilpotent linear part without the explicit knowledge of the triple. If one wishes one can compute the normal form such that it is guaranteed to lie in the kernel of an operator and one can be sure that this is really a normal form with respect to the nilpotent linear part; one can state that the normal form is in sl2-style. Although at first sight the normal form theory for maps is more complicated than for vector fields in the nilpotent case, it turns out that the final result is much better. Where in the vector field case one runs into invariant theoretical problems when the dimension gets larger if one wants to describe the general form of the normal form, for maps we obtain results without any restrictions on the dimension. In the literature only the two-dimensional nilpotent case has been described so far, as far as we know.

Original languageEnglish
Article number20210908
Pages (from-to)1-31
Number of pages31
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2261
Early online date4 May 2022
Publication statusPublished - 25 May 2022

Bibliographical note

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© 2022 Royal Society Publishing. All rights reserved.


  • dynamical system
  • nilpotent maps
  • normal form
  • sl2 representation


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