  We prove the convergence and ergodicity of a wide class of real and
higher-dimensional continued fraction algorithms, including folded and
$\alpha$-type variants of complex, quaternionic, octonionic, and Heisenberg
continued fractions, which we combine under the framework of Iwasawa continued
fractions. The proof is based on the interplay of continued fractions and
hyperbolic geometry, the ergodicity of geodesic flow in associated modular
manifolds, and a variation on the notion of geodesic coding that we refer to as
geodesic marking. As a corollary of our study of markable geodesics, we obtain
a generalization of Serret's tail-equivalence theorem for almost all points.
The results are new even in the case of complex continued fractions.
