We consider a space of complex polynomials of degree n ≥ 3 with n − 1 distinguished periodic orbits. We prove that the multipliers of these periodic orbits considered as algebraic functions on that space, are algebraically independent over C.

The multiplier λn of a periodic orbit of period n can be viewed as a (multiple-valued) algebraic function on the space of all complex quadratic polynomials pc(z) = z 2 + c. We provide a numerical algorithm for computing critical points of this function (i.e., points where the derivative of the multiplier with respect to the complex parameter c vanishes). We use this algorithm to compute critical points of λn up to period n = 10.

We consider the space of degree n ≥ 2 rational maps of the Riemann sphere with k distinct marked periodic orbits of given periods. First, we show that this space is irreducible. For k = 2n − 2 and with some mild restrictions on the periods of the marked periodic orbits, we show that the multipliers of these periodic orbits, considered as algebraic functions on the above mentioned space, are algebraically independent over C. Equivalently, this means that at its generic point, the moduli space of degree n rational maps can be locally parameterized by the multipliers of any 2n − 2 distinct periodic orbits, satisfying the above mentioned conditions on their periods. This work extends previous similar result obtained by the author for the case of complex polynomial maps.

We define an analytic setting for renormalization of unimodal maps with an arbitrary critical exponent. We prove the global Hyperbolicity of Renormalization conjecture for unimodal maps of bounded type with a critical exponent which is sufficiently close to an even integer.

The Kneser-Poulsen conjecture says that if a finite collection of balls in a Euclidean (spherical or hyperbolic) space is rearranged so that the distance between each pair of centers does not increase, then the volume of the union of these balls does not increase as well. We give new results about central sets of subsets of a Riemannian manifold and apply these results to prove new special cases of the Kneser-Poulsen conjecture in the twodimensional sphere and the hyperbolic plane.51M16 (51M25, 51M10, 53C20, 53C22) arXiv:1511.08134v3 [math.MG]