  We study geometrically finite one-dimensional mappings. These are a subspace
of $C^{1+\alpha}$ one-dimensional mappings with finitely many, critically
finite critical points. We study some geometric properties of a mapping in this
subspace. We prove that this subspace is closed under quasisymmetrical
conjugacy. We also prove that if two mappings in this subspace are
topologically conjugate, they are then quasisymmetrically conjugate. We show
some examples of geometrically finite one-dimensional mappings.
