  We establish a connection between the function space BMO and the theory of
quasisymmetric mappings on \emph{spaces of homogeneous type} $\widetilde{X}
:=(X,\rho,\mu)$. The connection is that the logarithm of the generalised
Jacobian of an $\eta$-quasisymmetric mapping $f: \widetilde{X} \rightarrow
\widetilde{X}$ is always in $\rm{BMO}(\widetilde{X})$. In the course of proving
this result, we first show that on $\widetilde{X}$, the logarithm of a
reverse-H\"{o}lder weight $w$ is in $\rm{BMO}(\widetilde{X})$, and that the
above-mentioned connection holds on metric measure spaces $\widehat{X}
:=(X,d,\mu)$. Furthermore, we construct a large class of spaces $(X,\rho,\mu)$
to which our results apply. Among the key ingredients of the proofs are
suitable generalisations to $(X,\rho,\mu)$ from the Euclidean or metric measure
space settings of the Calder\'{o}n--Zygmund decomposition, the Vitali Covering
Theorem, the Radon--Nikodym Theorem, a lemma which controls the distortion of
sets under an $\eta$-quasisymmetric mapping, and a result of Heinonen and
Koskela which shows that the volume derivative of an $\eta$-quasisymmetric
mapping is a reverse-H\"{o}lder weight.

Classical Analysis and ODEs