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By Solomon Lefschetz

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The monodromy mapping (or successor function) of equation (1) is defined to be the mapping of the y-axis onto itself which associatesto each point (0, y) the value when x = 27~of the solution of the equation with initial condition (0, y). We use the same term for the corresponding mapping of the circle lR/(27cZ). The monodromy mapping and its inverse are differentiable; it differs from the identity by a function a called the angular function: A(Y)=Y+4Y)> 4Y+274=4Y), a’(y)> - 1. (2) Definitions.

Topologically Unstabilizable Jets. First we choose a fixed coordinate system. An M-jet of a vector field is said to be an extension of an N-jet (M> N) if its Taylor polynomial of degree A4 can be obtained from the Taylor polynomial of the N-jet by adding terms of higher degree. An Invariant Definition. An M-jet of a vector field is an extension of an N-jet if M >N and the vector fields belonging to the M-jet belong to the set of fields representing the N-jet. Definition. A jet V’ (NN)of a vector field at a singular point is said to be topologically unstabilizable in the class of smooth (analytic) germs of vector fields if any higher order jet which is an extension of VN) contains smooth (analytic) representatives which are not topologically equivalent in any neighbourhood of the singular point.

There is a set of total measure on the real line such that any Cr diffeomorphism of the circle whose rotation number belongs to the set is C’-‘-smoothly equivalent to a rotation. Here, r is 00, o or any integer greater than two and, by convention, co -2= 00 and o--2=0 (where C” denotes the class of analytic mappings). There is a sufficient condition, known as Hermann’s “condition A”, on an irrational rotation number p for p to belong to the set mentioned in the theorem. To state it, let ~=a,+l/(a,+l/(a2+ .

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