Galois theory of periodic orbits of rational maps

The periodic points of a rational mapping are roots of a polynomial. If the coefficients of the mapping are algebraic numbers, then the periodic orbits are also algebraic numbers. A sequence of algebraic number fields is naturally associated with rational mappings, namely the fields containing all orbits of a given period. The authors study the corresponding Galois groups. They show that the latter have subgroups that permute the points of an orbit in the same way as the dynamics. The subgroup having all orbits as invariant sets identifies a field which contains the multipliers of the orbits. They construct their minimal polynomial, thereby computing the multiplier of a cycle without computing the cycle itself. They show that the periodic orbits of the quadratic family are soluble by radicals if their period is less or equal to 4, and they exhibit examples of unsoluble orbits of period 5. Dynamics over algebraic number fields is discrete, and all numerical experiments are reproducible.