Geometric and ergodic properties of certain classes of continued fractions

Continued fractions are systematically studied in number theory, dynamical systems, and ergodic theory, and appear frequently in other areas of mathematics. The first part of this thesis extends the connection between the dynamics of regular continued fractions and the geodesics on the modular surface $\operatorname{PSL}(2, \mathbb{Z})\backslash\mathbb{H}$ to the odd and grotesque continued fractions and the even continued fractions, as well as the Lehner and Farey expansions. I describe the natural extension of the corresponding Gauss maps as cross sections of the geodesic flow on modular surfaces. For the odd and grotesque continued fractions, $\Gamma$ is an index two subgroup of $\operatorname{PSL}(2,\Z)$; for the even continued fractions, $\Gamma$ is an index three subgroup; and for the Lehner and Farey expansions, $\Gamma=\operatorname{PSL}(2,\Z)$. Nakada’s $\alpha$-expansions interpolate between the regular continued fractions ($\alpha=1$), Hurwitz singular continued fractions ($\alpha=\frac{\sqrt{5}-1}{2}$), and nearest integer continued fractions ($\alpha=\frac{1}{2}$). The second part of this thesis introduces a new version of $\alpha$-expansions where all partial quotients are odd integers. I provide an explicit description of the natural extension of the corresponding Gauss map for $\frac{\sqrt{5}-1}{2}\leq\alpha\leq\frac{\sqrt{5}+1}{2}$, and investigate several of the ergodic properties of these maps