Orbital carriers and inheritance in discrete-time quadratic dynamics

TL;DR

This paper provides explicit formulas for orbital carriers of periods 4 to 6 in discrete quadratic dynamics and explores orbital inheritance up to period 12, revealing nested orbit structures that impact semiclassical trace formulas.

Contribution

It introduces systematic formulas for orbital carriers and investigates orbital inheritance, showing nested orbit structures in quadratic dynamics up to period 12.

Findings

Explicit formulas for periods 4, 5, 6 orbital carriers.

Orbital inheritance allows constructing unknown orbits from known ones.

Nested orbit stratification influences semiclassical trace formulas.

Abstract

Explicit formulas for {\sl orbital carriers} of periods $4$, $5$, and $6$ are reported for discrete-time quadratic dynamics. A systematic investigation of {\sl orbital inheritance} for periods as high as $k\leq 12$ is also reported. Inheritance means that unknown orbits may be obtained by nonlinear transformations of known orbits. Such nested {\sl orbit within orbit stratification} shows orbits not to be necessarily independent of each other as generally assumed. Orbital stratification is potentially significant to rearrange trajectories sums in trace formulas underlying modern semiclassical interpretations of atomic physics spectra. The stratification seems to dominate as the orbital period grows.