You're right that the g1 and g2 in section 2 are also scaled in addition to being shifts of f. I plan to clarify this in the text. This work is inspired by the

You're right that the g1 and g2 in section 2 are also scaled in addition to being shifts of f. I plan to clarify this in the text. This work is inspired by the self-similarity of fractals, which is in some cases primarily an intuitive or informal idea: for example, the Mandelbrot set is considered self-similar, but I'm not aware (despite searching) of any simple automorphisms of the Mandelbrot.