For regular polynomial endomorphisms of C^n and endomorphisms of P^n, any family of as many multipliers as the moduli dimension, from distinct cycles of period at least four (or five in one exceptional case), is algebraically independent.
Gorbovickis, Algebraic independence of multipliers of periodic orbits i n the space of rational maps of the Riemann sphere , Mosc
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Independence of multipliers in several variables complex dynamics
For regular polynomial endomorphisms of C^n and endomorphisms of P^n, any family of as many multipliers as the moduli dimension, from distinct cycles of period at least four (or five in one exceptional case), is algebraically independent.