We prove that badly approximable points on any analytic nondegenerate curve in ${\mathbb{R}}^n$ are an absolute winning set. This confirms a key conjecture in the area stated by Badziahin and Velani in 2014 that represents a far-reaching generalization of Davenport’s problem from the 1960s. Among various consequences of our main result is a solution to Bugeaud’s problem on real numbers badly approximable by algebraic numbers of arbitrary degree. The proof relies on new ideas from fractal geometry and homogeneous dynamics.

This paper investigates the structure of gravitational singularities at the level of the connection. We show in particular that for FLRW space-times with particle horizons a local holonomy, which is related to a gravitational energy, becomes unbounded near the big-bang singularity. This implies the $C_{\mathrm{loc}}^{0,1}$-inextendibility of such FLRW space-times. Again using an unbounded local holonomy, we also give a general theorem establishing the $C_{\mathrm{loc}}^{0,1}$-inextendibility of spherically symmetric weak null singularities which arise at the Cauchy horizon in the interior of black holes. Our theorem does not presuppose the mass-inflation scenario and in particular applies to the Reissner–Nordström-Vaidya space-times, as well as to space-times which arise from small and generic spherically symmetric perturbations of two-ended subextremal Reissner–Nordström initial data for the Einstein–Maxwell scalar field system. In previous work, Luk and Oh proved the $C^2$-formulation of strong cosmic censorship for this latter class of space-times—and based on their work we improve this to a $C_{\mathrm{loc}}^{0,1}$-formulation of strong cosmic censorship.

In this paper, we study the dynamics of degenerating sequences of rational maps on Riemann sphere $\hat{\mathbb{C}}$ using $\mathbb{R}$-trees. As an analogue of isometric group actions on $\mathbb{R}$-trees for Kleinian groups, we give two constructions for limiting dynamics on $\mathbb{R}$-trees: one geometric and one algebraic. The geometric construction uses the limit of rescalings of barycentric extensions of rational maps, while the algebraic construction uses the Berkovich space of complexified Robinson’s field. We show that the two approaches are equivalent. As an application, we use it to give a classification of hyperbolic components of rational maps that admit degeneracies with bounded multipliers.

We prove a new integrability principle for gradient variational problems in ${\mathbb{R}}^2$, showing that solutions are explicitly parameterized by κ-harmonic functions, that is, functions which are harmonic for the Laplacian with varying conductivity κ, where κ is the square root of the Hessian determinant of the surface tension.