Determinacy of sufficiently differentiable maps

Selby, Alan M.

doi:10.1090/S0002-9947-1989-0937251-3

Determinacy of sufficiently differentiable maps
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by Alan M. Selby

Trans. Amer. Math. Soc. 312 (1989), 85-113

DOI: https://doi.org/10.1090/S0002-9947-1989-0937251-3

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Abstract:

Variants of the algebraic conditions of Mather are shown to be sufficient for the $k$-determinacy of ${C^u}$ maps with respect to $j$-flat, contact (or right) ${C^r}$ equivalence relations where $u - k \leq r \leq u - k + j + 1$ and $0 \leq j < k \leq u$. The required changes of coordinates and matrix-valued functions are constructed from the variation of coefficients in polynomials. The main result follows from a finite-dimensional, polynomial pertubation argument which employs a parameter-dependent polynomial representation of functions based on Taylor’s formula. For $r > k$, the algebraic conditions are seen to be necessary.

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