Nonlinear dependencies between sets of periodic orbits

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Abstract
We report the existence of a phenomenon of inheritance among periodic trajec- tories of the quadratic map. Within the set Sk of all possible trajectories of period k we flnd mother" trajectories from which several daughters" may be derived by simple polynomial transformations. For example, from the six orbital points zi of a period-six mother we get three additional period-six daughters as the zeros of the six cubics x3 ¡3x¡zi = 0. Daughters might have daughters. This stratiflcation shows that periodic orbits are not necessarily independent of each other. This fact could be of importance for decomposing certain sums involving sets of periodic trajectories, particularly for trace formulas underlying semiclassical interpretations of spectra in atomic physics.