One-dimensional polynomial maps, periodic points and multipliers (Q2854094)

The paper considers multipliers of periodic orbits for complex polynomial dynamical systems in one variable, addressing in particular the behavior of such multipliers under perturbation of a polynomial map. The main results show that the multipliers of a set of distinct periodic orbits of a polynomial map on \(\mathbb{C}\) will in general vary independently if the coefficients of the polynomial are perturbed.NEWLINENEWLINEGiven a monic polynomial \(g(z)\) of degree \(n\) and points \(\beta_1,\dots,\beta_l\) in \(\mathbb{C}\) with exact periods \(r_1,\dots,r_l\) under \(g(z)\), the behavior of the orbit of each \(\beta_i\) is understood via the map \(\text{Mult}_{\beta_i}: f(z) \mapsto (f^{\circ r_i})'(\beta_i(f))\) whose domain is a neighborhood (in the space of monic polynomials of degree \(n\)) of \(g(z)\) defined so that each \(\beta_i\) can be viewed as a holomorphic function on the neighborhood such that \(\beta_i(f)\) has exact period \(r_i\) under \(f(z)\). So the multipliers of the orbits of \(\beta_1,\dots,\beta_l\) vary independently at \(g(z)\) if the gradients of \(\text{Mult}_{\beta_1},\dots,\text{Mult}_{\beta_l}\) at \(g(z)\) are linearly independent in \(\mathbb{C}^n\). The paper concludes that independence is achieved for a generic choice of \(g(z)\) if \(n \geq 3\), \(r_1,\dots,r_l < n\) (with equality allowed if no \(\beta_i\) is a fixed point), and the orbits of \(\beta_1,\dots,\beta_l\) are pairwise disjoint. The conclusion follows from computations of the gradients of \(\text{Mult}_{\beta_1},\dots,\text{Mult}_{\beta_l}\) which show that maps of the form \(z\mapsto z^n\) achieve independence under the given assumptions, combined with an argument that the set of monic polynomial maps achieving linear independence must be Zariski open.