Abstract

Suppose T : R2 --f R2 is a real analytic mapping defined by the series representation y1 = cx + dy + *.* (1.1) converging in 0 < ) x / + / y / < 6. A curve y = G(X) is said to be an invariant curve of T if yi = #(x1). Such curves are important in the theory of periodic stability of dynamical systems. From this viewpoint, the most significant class of maps is that of the area-preserving, whose invariants were studied by G. D. Birkhoff ([I], [2]). Although Birkhoff derived formal series expansions for the invariants, he was unable to show that these converge in all cases. J. Moser ([5], [6]) determined invariants of area-preserving transformations near a hyperbolic fixpoint. The book of P. Monte1 ([4]) contains an account of the classical work of Hadamard, Lattes and Poincare. By the use of majorization techniques, these analysts showed that invariant curves of an analytic nonsingular transformation exist, provided one of the eigenvalues, o, 7, of the matrix (z i) has modulus not equal to one, and u # rk, d f T, k = 2, 3 ,... . Invariant curves are of some significance in the theory of fractional itera- tion. In particular, if (z i) is similar (under some homogeneous linear trans- formation) to (i i), the invariants play a major role in the construction of the iterates (see [3]). Henceforth it is assumed that the series representation of I is of the form

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