  The gravitational dynamics and cosmological implications of three classes of
recently introduced multi-scale spacetimes (with, respectively, ordinary,
weighted and q-derivatives) are discussed. These spacetimes are non-Riemannian:
the metric structure is accompanied by an independent measure-differential
structure with the characteristics of a multi-fractal, namely, different
dimensionality at different scales and, at ultra-short distances, a discrete
symmetry known as discrete scale invariance. Under this minimal paradigm, five
general features arise: (a) the big-bang singularity can be replaced by a
finite bounce, (b) the cosmological constant problem is reinterpreted, since
accelerating phases can be mimicked by the change of geometry with the time
scale, without invoking a slowly rolling scalar field, (c) the discreteness of
geometry at Planckian scales can leave an observable imprint of logarithmic
oscillations in cosmological spectra and (d) give rise to an alternative
mechanism to inflation or (e) to a fully analytic model of cyclic mild
inflation, where near scale invariance of the perturbation spectrum can be
produced without strong acceleration. Various properties of the models and
exact dynamical solutions are discussed. In particular, the multi-scale
geometry with weighted derivatives is shown to be a Weyl integrable spacetime.
