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Under the AdS/CFT correspondence, asymptotically AdS geometries withbackreaction can be viewed as CFT states subject to a renormalization group(RG) flow from an ultraviolet (UV) description towards an infrared (IR) sector.For black holes however, the IR point is the horizon, so one way to interpretthe interior is as an analytic continuation to a "trans-IR" imaginary-energyregime. In this paper, we demonstrate that this analytic continuation preservessome imprints of the UV physics, particularly near its "endpoint" at theclassical singularity. We focus on holographic phase transitions of geometricobjects in round black holes. We first assert the consistency of interpretingsuch black holes, including their interiors, as RG flows by constructing amonotonic $a$-function. We then explore how UV phase transitions ofentanglement entropy and scalar two-point functions, each of which are encodedby bulk geometry under the holographic mapping, are connected to the structureof the near-singularity geometry, which is characterized by Kasner exponents.Using 2d holographic flows triggered by relevant scalar deformations as testbeds, we find that the 3d bulk's near-singularity Kasner exponents can beviewed as functions of the UV physics precisely when the deformation isnonzero.

Imprints of phase transitions on Kasner singularities