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DTSTAMP:20261105T090223
DTSTART;TZID=America/Detroit:20261109T160000
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SUMMARY:Lecture / Discussion:Dynamics seminar: Dynamics of the Secant Method and Other Monomial-Type Maps
DESCRIPTION: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.\n\nAdditionally\, 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). \n\nBy 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.
UID:153015-21914925@events.umich.edu
URL:https://events.umich.edu/event/153015
CLASS:PUBLIC
STATUS:CONFIRMED
CATEGORIES:Dynamics
LOCATION:East Hall - 3096
CONTACT:
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