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I hope to illustrate in these lectures that the central ideas of the theory are, in fact, quite simple. The nondegeneracy condition on the integrable approximation is somewhat different for both cases. This proof essentially follows the traditional lines laid out by the inventors of this theory, kolmogorov, arnold and moser (whence the acronym ‘kam’), and the emphasis is more on the underlying ideas than on the sharpness of the arguments.

The goal is to present the background and to explain and compare somewhat informally some of the main methods of proof Around 1980, i came back to dynamical systems, and i quickly realized that the kam theorem. It is an expanded version of the lectures given by the author in the summer research institute on smooth ergodic theory seattle, 1999.

The remarkable discovery of kam theory was that even though a solution of (1.2) cannot be expected for rational frequency vectors it does indeed exist if the frequencies !1

!n are rationally independent and, moreover, satisfy the diophantine inequalities By thomas ransford i first heard about the kam theorem when i was an undergra It seemed to me the most beautiful result in the world, but for many years my interests were

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