Local homeomorphy of some mappings with bounded distortion and quasiconformality coefficient less than two (Q1921788)

Suppose that \(f:D \to\mathbb{R}^n\) is a nonconstant quasiregular mapping and that \(a\in E\). Set \(L_f(a,t) = \max |f(x)-f(a)|\) for \(|x-a|=t\). Now \(f\) is said to be locally homogeneous at \(a\) if the limit \(\lim_{t\to 0} |f(a+tx) - f(a)|/L_f (a,t)\) exists. The author proves, using a capacity argument, that if \(f\) is locally homogeneous at each point of \(D\), then the inner dilatation \(K_I(f)\) of \(f\) satisfies \(K_I(f)\geq 2\) provided that the branch set of \(f\) is nonempty.