Geometry and Algebra of the Deltoid Map

The geometry of the deltoid curve gives rise to a self-map of ${\mathbb{C}}^2$ that is expressed in coordinates by $f(x,y)=(y^2-2x,x^2-2y)$. This is one in a family of maps that generalize Chebyshev polynomials to several variables. We use this example to illustrate two important objects in complex dynamics: the Julia set and the iterated monodromy group.