  We constructed a model of natural inflation in the context of
$\alpha$-attractor supergravity, in which both the dilaton field and the axion
field are light during inflation, and the inflaton may be a combination of the
two. The T-model version of this theory is defined on the Poincare disk with
radius |Z| = 1. It describes a Mexican hat potential with the flat axion
direction corresponding to a circle of radius |Z| < 1. The axion decay constant
$f_{a}$ in this theory can be exponentially large because of the hyperbolic
geometry of the Poincare disk. Depending on initial conditions, this model may
describe $\alpha$-attractor inflation driven by the radial component of the
inflaton field, natural inflation driven by the axion field, or a sequence of
these two regimes. We also construct the E-model version of this theory, which
has similar properties. In addition, we describe generalized $\alpha$-attractor
models where the potential can be singular at the boundary of the moduli space,
and show that they can provide a simple solution for the problem of initial
conditions for the models with plateau potentials.
