Dynamics seminar: Dynamics of the Secant Method and Other Monomial-Type Maps

We discuss the secant method, a root-finding algorithm alternative to Newton’s method. Given a rational map f : P^1 -> P^1, the secant method yields a dynamical system Sf on P^1 x P^1 (minus a finite indeterminacy locus). If r in C is a simple root of f, then the point (r, r) is an attracting fixed point of Sf.

Additionally, when 0 is a simple root of f with f’’(0) nonzero, we have that Sf is an instance of the broader class of “monomial-type maps.” A “monomial map” is simply a function whose output coordinates are all monomials in the input coordinates. A “monomial-type map” is then a coordinate-by-coordinate scaling of a monomial map by some holomorphic scalar functions. In the aforementioned case for 0 and Sf, we find that Sf(x, y) = (a(x, y) * xy, x).

By extending the 1-dimensional constructions of Böttcher coordinates and Green’s functions to these higher-dimensional monomial-type maps, we can show that (1) Sf has well-defined dynamical potential, (2) it takes the vequipotential to the v^φ equipotential (where φ is the golden ratio), and (3) Sf is locally modeled by the Fibonacci monomial map (x, y) |-> (xy, x), whose dynamics are well-understood.