Normal Forms of Maps: Formal and Algebraic Aspects

Gramchev, Todor; Walcher, Sebastian

doi:10.1007/s10440-005-1140-2

Cited by 36 publications

(9 citation statements)

References 18 publications

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“…Lewis [10] proved that if a transformation u ∈ GS n (K) satisfies so-called pseudo-incommensurable condition, then the iteration problem has a formal power series solution. This Lewis result has been repeatedly proved by different authors [5,6,9].…”

Section: Introductionsupporting

confidence: 56%

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Iterations and Groups of Formal Transformations

Капцов¹

2021

J. Sib. Fed. Univ., Math. phys.

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Section: Introductionsupporting

confidence: 56%

“…The iteration problem was investigated by Koenigs, Lewis, Baker, Chen, Sternberg and others. Bibliographical references can be found in [4][5][6]. Every smooth flow f t is defined by a system of ordinary differential equations…”

Section: Introductionmentioning

confidence: 99%

Iterations and Groups of Formal Transformations

Капцов¹

2021

J. Sib. Fed. Univ., Math. phys.

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“…Lewis [10] proved that if a transformation u ∈ GS n (K) satisfies so-called pseudoincommensurable condition, then the iteration problem has a formal power series solution. This Lewis result has been repeatedly proved by different authors [5,6,9].…”

supporting

confidence: 56%

Iterations and groups of formal transformations

Kaptsov¹

2020

Preprint

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“…There are few papers regarding the study of normal forms for maps [4][5][6][7], and there seems to be no general and systematic approach to the construction of normal forms of maps. However, the recent appearance of [8] confirms that there is indeed a need for the theoretical development of normal form theory for maps.…”

Section: Introductionmentioning

confidence: 99%

Normal form for maps with nilpotent linear part

2022

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The normal form for an n -dimensional map with irreducible nilpotent linear part is determined using s l 2 -representation theory. We sketch by example how the reducible case can also be treated in an algorithmic manner. The construction (and proof) of the s l 2 -triple from the nilpotent linear part is more complicated than one would hope for, but once the abstract s l 2 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 s l 2 -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.

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Normal Forms of Maps: Formal and Algebraic Aspects

Cited by 36 publications

References 18 publications

Iterations and Groups of Formal Transformations

Iterations and Groups of Formal Transformations

Iterations and groups of formal transformations

Normal form for maps with nilpotent linear part

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