Dynamics Seminar

In this talk, I will show that the multipliers at the cycles with periods 1 and 2 provide a good description of the space P_d of polynomial maps of degree d modulo conjugation by affine transformations. More precisely, the elementary symmetric functions of the multipliers at the cycles with periods 1 and 2 induce a finite birational morphism from P_d onto its image. This result arises as a direct consequence of the following two facts: (1) For each integer p > 1, any sequence of complex polynomials of degree d with bounded multipliers at its cycles with period p is necessarily bounded in P_d(C). (2) A generic conjugacy class of complex polynomials of degree d is uniquely determined by its multipliers at its cycles with periods 1 and 2. I will present a quantitative version of the first statement, which also holds over various valued fields of characteristic 0. The second statement proves a conjecture by Hutz and Tepper and strengthens a recent result by Ji and Xie in the polynomial case.