Homogenization of a quasimorphism

Definition
Suppose $$f:G \to \R$$ is a defining ingredient::quasimorphism. The homogenization of $$f$$ is a defining ingredient::homogeneous quasimorphism $$\mu_f$$ (sometimes denoted $$\overline{f}$$) defined as follows:

$$\mu_f := x \mapsto \lim_{n \to \infty} \frac{f(x^n)}{n}$$

Defect

 * Upper bound: The defect $$D(\mu_f)$$ of the homogenization is at most twice the defect $$D(f)$$ of $$f$$.
 * Lower bound: There is no effective lower bound besides the obvious one of zero. To see this, note that any quasimorphism with bounded image has homogenization the zero map, even though we can have quasimorphisms with bounded image that have arbitrarily large defect.