The Mandelbrot set and $\sigma$-automorphisms of quotients of the shift
Pau Atela · Transactions of the American Mathematical Society · 1993
In this paper we study how certain loops in the parameter space of quadratic complex polynomials give rise to shift-automorphisms of quotients of the set ${\Sigma _2}$ of sequences on two symbols. The Mandelbrot set ${\mathbf {M}}$ is the set of parameter values for which the Julia set of the corresponding polynomial is connected. Blanchard, Devaney, and Keen have shown that closed loops in the complement of the Mandelbrot set give rise to shift-automorphisms of ${\Sigma _2}$ , i.e., homeomorphisms of ${\Sigma _2}$ that commute with the shift map. We study what happens when the loops are not entirely in the complement of the Mandelbrot set. We consider closed loops that cross the Mandelbrot set at a single main bifurcation point, surrounding a component of ${\mathbf {M}}$ attached to the main cardioid. If $n$ is the period of this component, we identify a period- $n$ orbit of ${\Sigma _2}$ to a single point. The loop determines a shift-automorphism of this quotient space of ${\Sigma _2}$ . We give these maps explicitly.