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.
Bousch, Sur quelques probl´ emes de la dynamique holomorphe, 1992
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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.