  In this paper, we introduce a class of homeomorphisms between metric spaces,
which are locally biH\"{o}lder continuous mappings. Then an embedding result
between Besov spaces induced by locally biH\"{o}lder continuous mappings
between Ahlfors regular spaces is established, which extends the corresponding
result of Bj\"{o}rn-Bj\"{o}rn-Gill-Shanmugalingam (J. Reine Angew. Math. 725:
63-114, 2017). Furthermore, an example is constructed to show that our
embedding result is more general. We also introduce a geometric condition,
named as uniform boundedness, to characterize when a quasisymmetric mapping
between uniformly perfect spaces is locally biH\"{o}lder continuous.
