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We elaborate on the anholonomic frame deformation method, AFDM, forconstructing exact solutions with quasiperiodic structure in modified gravitytheories, MGTs, and general relativity, GR. Such solutions are described bygeneric off-diagonal metrics, nonlinear and linear connections and (effective)matter sources with coefficients depending on all spacetime coordinates viacorresponding classes of generation and integration functions and (effective)matter sources. There are studied effective free energy functionals andnonlinear evolution equations for generating off-diagonal quasiperiodicdeformations of black hole and/or homogeneous cosmological metrics. Thephysical data for such functionals are stated by different values of constantsand prescribed symmetries for defining quasiperiodic structures at cosmologicalscales, or astrophysical objects in nontrivial gravitational backgrounds somesimilar forms as in condensed matter physics. It is shown how quasiperiodicstructures determined by general nonlinear, or additive, functionals forgenerating functions and (effective) sources may transform black hole likeconfigurations into cosmological metrics and inversely. We speculate onpossible implications of quasiperiodic solutions in dark energy and dark matterphysics. Finally, it is concluded that geometric methods for constructing exactsolutions consist an important alternative tool to numerical relativity forinvestigating nonlinear effects in astrophysics and cosmology.

Deforming Black Hole and Cosmological Solutions by Quasiperiodic and/or Pattern Forming Structures in Modified and Einstein Gravity