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1. Bounded Geometry and p-Harmonic Functions Under Uniformization and Hyperbolization

   https://doi.org/10.1007/s12220-020-00477-0  

   Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari (May 2021, The Journal of Geometric Analysis)

   null (Ed.)

   Abstract The uniformization and hyperbolization transformations formulated by Bonk et al. in “Uniformizing Gromov Hyperbolic Spaces” , Astérisque, vol 270 (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly starlike Gromov hyperbolic space is equipped with a measure that is uniformly locally doubling and supports a uniformly local p -Poincaré inequality, then the transformed measure is globally doubling and supports a global p -Poincaré inequality on the corresponding uniformized space. In the opposite direction, we show that such global properties on bounded locally compact uniform spaces yield similar uniformly local properties for the transformed measures on the corresponding hyperbolized spaces. We use the above results on uniformization of measures to characterize when a Gromov hyperbolic space, equipped with a uniformly locally doubling measure supporting a uniformly local p -Poincaré inequality, carries nonconstant globally defined p -harmonic functions with finite p -energy. We also study some geometric properties of Gromov hyperbolic and uniform spaces. While the Cartesian product of two Gromov hyperbolic spaces need not be Gromov hyperbolic, we construct an indirect product of such spaces that does result in a Gromov hyperbolic space. This is done by first showing that the Cartesian product of two bounded uniform domains is a uniform domain. 

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2. Continuous Regular Functions

   https://doi.org/https://doi.org/10.23638/LMCS-16(1:17)2020  

   Block Gorman, Alexi Block; Hieronymi, Philipp; Kaplan, Elliot; Meng, Ruoyu; Walsberg, Erik; Wang, Zihe; Xiong, Ziqin; Yang, Hongru (January 2020, Logical methods in computer science)

   Following Chaudhuri, Sankaranarayanan, and Vardi, we say that a function f:[0,1]→[0,1] is r-regular if there is a Büchi automaton that accepts precisely the set of base r∈N representations of elements of the graph of f. We show that a continuous r-regular function f is locally affine away from a nowhere dense, Lebesgue null, subset of [0,1]. As a corollary we establish that every differentiable r-regular function is affine. It follows that checking whether an r-regular function is differentiable is in PSPACE. Our proofs rely crucially on connections between automata theory and metric geometry developed by Charlier, Leroy, and Rigo. 

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3. Frames and Finite-Rank Integral Representations of Positive Operator-Valued Measures

   https://doi.org/10.1007/s10440-019-00252-6  

   Gabardo, Jean-Pierre; Han, Deguang (April 2019, Acta Applicandae Mathematicae)

   Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) that have integral representations using rank-one operators. However, not every POVM has an integral representation. One goal of this paper is to examine the POVMs that have finite-rank integral representations. More precisely, we present a necessary and sufficient condition under which a positive operator-valued measure $$F: \Omega \to B(H)$$ has an integral representation of the form $$F(E) =\sum_{k=1}^{m} \int_{E}\, G_{k}(\omega)\otimes G_{k}(\omega) d\mu(\omega)$$ for some weakly measurable maps $$G_{k} \ (1\leq k\leq m) $$ from a measurable space $$\Omega$$ to a Hilbert space $$\mathcal{H}$$ and some positive measure $$\mu$$ on $$\Omega$$. Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic. 

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4. CscK metrics near the canonical class

   https://doi.org/10.1515/crelle-2024-0096  

   Guo, Bin; Jian, Wangjian; Shi, Yalong; Song, Jian (January 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let 𝑋 be a Kähler manifold with semiample canonical bundle $K_X$K_{X}.It is proved in [W. Jian, Y. Shi and J. Song, A remark on constant scalar curvature Kähler metrics on minimal models,Proc. Amer. Math. Soc.147(2019), 8, 3507–3513] that, for any Kähler class 𝛾, there exists $\delta >0$\delta>0such that, for all $t\in (0,\delta )$t\in(0,\delta), there exists a unique cscK metric $g_t$g_{t}in $K_X+t\gamma$K_{X}+t\gamma.In this paper, we prove that ${\left\{(X,g_t)\right\}}_{t\in (0,\delta )}$\{(X,g_{t})\}_{t\in(0,\delta)}have uniformly bounded Kähler potentials, volume forms and diameters.As a consequence, these metric spaces are pre-compact in the Gromov–Hausdorff sense. 

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5. Schrödinger operators with potentials generated by hyperbolic transformations: I—positivity of the Lyapunov exponent

   https://doi.org/10.1007/s00222-022-01157-2  

   Avila, Artur; Damanik, David; Zhang, Zhenghe (February 2023, Inventiones mathematicae)

   Abstract We consider discrete one-dimensional Schrödinger operators whose potentials are generated by sampling along the orbits of a general hyperbolic transformation. Specifically, we show that if the sampling function is a non-constant Hölder continuous function defined on a subshift of finite type with a fully supported ergodic measure admitting a local product structure and a fixed point, then the Lyapunov exponent is positive away from a discrete set of energies. Moreover, for sampling functions in a residual subset of the space of Hölder continuous functions, the Lyapunov exponent is positive everywhere. If we consider locally constant or globally fiber bunched sampling functions, then the Lyapuonv exponent is positive away from a finite set. Moreover, for sampling functions in an open and dense subset of the space in question, the Lyapunov exponent is uniformly positive. Our results can be applied to any subshift of finite type with ergodic measures that are equilibrium states of Hölder continuous potentials. In particular, we apply our results to Schrödinger operators defined over expanding maps on the unit circle, hyperbolic automorphisms of a finite-dimensional torus, and Markov chains. 

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