Dynamical symmetries

Abstract

This chapter studies various notions of dynamical symmetries for polynomials. It begins by defining the group of dynamical symmetries Σ‎(P) of a single polynomial P and present various characterizations of it especially in the Archimedean case. The chapter investigates the variation of this symmetry group when P belongs to an algebraic family. It then proves that Σ‎(P) consists of those affine transformations mapping the set of preperiodic points of P onto itself. The chapter also introduces the notion of primitive polynomials, which are polynomials that cannot be written as iterates of polynomials of lower degree up to symmetries. It was progressively realized that a polynomial might have symmetries induced by polynomials of degree ≥ 2. The chapter investigates this phenomenon, building on Ritt's theory. It concludes by describing the stratification of the moduli space of polynomials in degree d ≤ 6 induced by the presence of symmetries.