Let $M_n$ be the moduli space of degree $n$ rational or polynomial maps of the Riemann sphere and let $d_n$ be its dimension.

In the first part of the talk we show that the multipliers of any $d_n$ distinct periodic orbits (with very mild restrictions on the periods) considered as multiple-valued algebraic maps on $M_n$, are not related by any algebraic relations, hence, provide a local parameterization of $M_n$ in a neighborhood of its generic point. It is then a natural question to describe the set of maps in $M_n$ at which this local parameterization fails, that is, to describe the set of all critical points of the multiplier map, defined as the map which assigns to each function in $M_n$ the $d_n$-tuple of its multipliers at the chosen periodic orbits.

In the second half of the talk we address this question in the simplest possible case - the case of the quadratic family $z^2+c$. We show that as the period of the periodic orbits increases to infinity, the critical points of the multiplier map equidistribute on the boundary of the Mandelbrot set.