Abstract
The infinite set of periodic orbits of a chaotic system is investigated. Their globally exact symbolic dynamics is related to Birkhoff's method of iterated symmetry lines which produces symmetric periodic orbits of various classes. A complete picture emerges as to what part of the periodic orbits can be obtained by means of the Birkhoff lines. In addition, the thermodynamic formalism is applied to the multifractal structure created by the horseshoe arrangement of the periodic orbits.
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