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Symbolic Dynamics and the Discrete Variational Principle

By Dullin Department Of and H. R. Dullin


We show how to construct symbolic dynamics for the class of 2d-dimensional twist mappings generated by piecewise strictly convex/concave generating functions. The method is constructive and gives an efficient way to find all periodic orbits of these high dimensional symplectic mappings. It is illustrated with the cardioid and the stadium billiard. 05.45.+b, 03.20.+i 1 Introduction By a dictum of Poincar'e, periodic orbits are "the only breach through which we may attempt to penetrate an area hitherto deemed inaccessible" [1, Vol. 1, x36]. Today this is a well established fact in the theory of dynamical systems. Classical global characteristic properties like Lyapunov exponents or diffusion coefficients are calculated by sums over periodic orbits [2, 3]. Particularly interesting is the fact that also quantum mechan1 ical properties can be revealed by summation over classical periodic orbits [4, 5]. This stimulated renewed interest in classifying and calculating periodic orbits [6..

Year: 2007
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