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Algebraic independence of local conjugacies and related questions in polynomial dynamics

Let $K$ be an algebraically closed field of characteristic 0 and $f\in K[t]$ a polynomial of degree $d\geq 2$. There exists a local conjugacy $ψ_f(t)\in tK[[1/t]]$ such that $ψ_f(t^d)=f(ψ_f(t))$. It has been known that $ψ_f$ is transcendental over $K(t)$ if $f$ is not conjugate to $t^d$ or a constant multiple of the Chebyshev polynomial. In this paper, we study the algebraic independence of $ψ_{f_1}$,\ldots,$ψ_{f_n}$ using a recent result of Medvedev-Scanlon. Related questions in transcendental number theory and canonical heights in arithmetic dynamics are also discussed.