Equicontinuous classes of ring $Q$-homeomorphisms

We give a description of ring $Q$-homeomorphisms in $\mathbb R^n$, $n\geqslant2$, and find a series of conditions for normality of families of ring $Q$-homeomorphisms. For a family to be normal it is sufficient that the dominant $Q(x)$ have logarithmic-type singularities of order at most $n-1$. Another sufficient condition for normality is that $Q(x)$ has finite mean oscillation at each point; for example, $Q(x)$ has finite mean value over infinitesimal balls. The definition of ring $Q$-homeomorphism is motivated by the ring definition of Gehring for quasiconformality. In particular, the mappings with finite length distortion satisfy a capacity inequality that justifies the definition of ring $Q$-homeomorphism. Therefore, deriving consequences of the theory to be presented, we obtain criteria for normality of families of homeomorphisms f with finite length distortion and homeomorphisms of the Sobolev class $W^{1,n}_\mathrm{loc}$ in terms of the inner dilation $K_I(x,f)$. Moreover, the class of strong ring $Q$-homeomorphisms for a locally summable $Q$ is closed.