  Long standing themes in inflation include the issue of large field vs. small
field inflation as well as the question what fraction of phase space leads to
sufficient inflation, and furthermore is compatible with the experimental data.
In the present paper these issues are discussed in the context of modular
inflation, a specialization of the framework of automorphic nonlinear
$\sigma$-models associated to homogeneous spaces $G/K$ in which the continuous
shift symmetry group $G$ is weakly broken to discrete subgroups $\Gamma$. The
target spaces of these theories inherit a curved structure from the group $G$,
which in the case of modular invariant inflation leads to a hyperbolic field
space geometry. It is shown that in this class of models the symmetry structure
leads to both large and small field inflationary trajectories within a single
modular inflation model. The present paper analyzes the concrete model of
$j$-inflation, a hyperbolic model with nontrivial inflaton interactions. It
describes in some detail the structure of the initial conditions, including a
systematic analysis of several phenomenological functions on the target space,
leading to constraints on the curvature scalar of the field space by upcoming
experiments, as well as a discussion of the scaling behavior of the spectral
index, the finite volume fraction of the field space leading to sufficient
inflation, the attractor behavior of $j$-inflation, and a comparison of
inflaton trajectories vs. target space geodesics. The tensor-ratio analysis
shows that $j$-inflation is an interesting target for upcoming ground and
satellite experiments.
