Real analytic open maps
P. T. Church, J. G. Timourian · Pacific Journal of Mathematics · 1974
Let R and C be the real and complex fields, respectively, and for ζeC let ^(ζ) be the real part of ζ.If /: M» +1 -> N* is real analytic and open with p Ξ> 1, then there is a closed subspace XdM p+1 such that dim/(X) ^ p -2 and, for every x 6 M p+1 -Xy there is a natural number d(x) with / at x locally topologically equivalent to the map φ M : C x R*-1 -> R x R*~ι defined by φ dM {z 9 t l9 , t p ^) = (^(z*), t lf , ί^) .In [7] Nathan proved: It f: M 2 -> N 1 is real analytic and open, then for every xeM 2 there is a natural number d(x) with / at x locally topologically equivalent to the map φ d(x) : C -> R defined by Φπ*)(z) -&(z dix) )> This is the case p = 1 of the above theorem, but our proof is not a generalization of his.Examples (see (3.3)) show that "topologically equivalent" cannot be replaced by "analytically equivalent" or even "C 1 equivalent", / real analytic cannot be replaced by / C°° (but see (3.1)), an exceptional set X with dim/(X) ^ p -2 is needed, and dim X may be p-1.