Analytical study of Cantori: Gap structure, initial conditions, and dimensions

H. J. Schellnhuber and H. Urbschat
Phys. Rev. A 38, 5888 – Published 1 December 1988
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Abstract

A complete exact analysis of invariant Cantor sets in a piecewise linear but nontrivial area-preserving twist map is given. The results are based on concise formulas for minimizing periodic orbits, which in turn are derived from rigorous work by Aubry and Percival using only fundamental relations of continued-fraction theory. The exponential law for the decay of Cantori gaps, however, is found from the mapping itself with the aid of an elementary argument. The general scheme determining the initial conditions is transformed in the case of noble Cantori to an equivalent dynamical system permitting superexponentially fast calculations. By constructing polygonal embeddings the fractal and Hausdorff dimension of the invariant sets considered is proved to be precisely zero under all circumstances. Thus most notions and generic properties of Cantori can be understood within the frame of an explicitly solved example.

  • Received 11 July 1988

DOI:https://doi.org/10.1103/PhysRevA.38.5888

©1988 American Physical Society

Authors & Affiliations

H. J. Schellnhuber and H. Urbschat

  • Fachbereich Physik, Universität Oldenburg, Postfach 25 03, D-2900 Oldenburg, Federal Republic of Germany

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Issue

Vol. 38, Iss. 11 — December 1988

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