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1. Uniformly branching trees

   https://doi.org/10.1090/tran/8404  

   Bonk, Mario; Meyer, Daniel (April 2022, Transactions of the American Mathematical Society)

   A quasiconformal tree T is a (compact) metric tree that is doubling and of bounded turning. We call T trivalent if every branch point of T has exactly three branches. If the set of branch points is uniformly relatively separated and uniformly relatively dense, we say that T is uniformly branching. We prove that a metric space T is quasisymmetrically equivalent to the continuum self-similar tree if and only if it is a trivalent quasiconformal tree that is uniformly branching. In particular, any two trees of this type are quasisymmetrically equivalent. 

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2. Decomposing multitwists

   https://doi.org/10.1007/s11854-023-0301-4  

   Fletcher, Alastair N; Vellis, Vyron (April 2024, Journal d'Analyse Mathématique)

   The Decomposition Problem in the class $$LIP(\S^2)$$ is to decompose any bi-Lipschitz map $$f:\S^2 \to \S^2$$ as a composition of finitely many maps of arbitrarily small isometric distortion. In this paper, we construct a decomposition for certain bi-Lipschitz maps which spiral around every point of a Cantor set $$X$$ of Assouad dimension strictly smaller than one. These maps are constructed by considering a collection of Dehn twists on the Riemann surface $$\S^2 \setminus X$$. The decomposition is then obtained via a bi-Lipschitz path which simultaneously unwinds these Dehn twists. As part of our construction, we also show that $$X \subset \S^2$$ is uniformly disconnected if and only if the Riemann surface $$\S^2 \setminus X$$ has a pants decomposition whose cuffs have hyperbolic length uniformly bounded above, which may be of independent interest. 

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3. A comparison principle for semilinear Hamilton–Jacobi–Bellman equations in the Wasserstein space

   https://doi.org/10.1007/s00526-024-02718-4  

   Daudin, Samuel; Seeger, Benjamin (April 2024, Calculus of Variations and Partial Differential Equations)

   Abstract The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton–Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the 2-Wasserstein distance in the doubling of variables argument, which is done by introducing a further entropy penalization that ensures that the relevant optima are achieved at positive, Lipschitz continuous densities with finite Fischer information. This allows to prove uniqueness and stability of viscosity solutions in the class of bounded Lipschitz continuous (with respect to the 1-Wasserstein distance) functions. The result does not appeal to a mean field control formulation of the equation, and, as such, applies to equations with nonconvex Hamiltonians and measure-dependent volatility. For convex Hamiltonians that derive from a potential, we prove that the value function associated with a suitable mean-field optimal control problem with nondegenerate idiosyncratic noise is indeed the unique viscosity solution. 

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4. Bi-Lipschitz arcs in metric spaces with controlled geometry

   https://doi.org/10.4171/RMI/1484  

   Honeycutt, Jacob; Vellis, Vyron; Zimmerman, Scott (May 2024, Revista Matemática Iberoamericana)

   In this paper, we generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincaré inequality). In particular, we find sharp conditions on metric measure spaces X so that any bi-Lipschitz embedding of a subset of the real line into X extends to a bi-Lipschitz embedding of the whole line. Along the way, we prove that if the complement of an open subset Y of X has small Assouad dimension, then it is a uniform domain. Finally, we prove a quantitative approximation of continua in X by bi-Lipschitz curves. 

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5. Conformal Transformation of Uniform Domains Under Weights That Depend on Distance to The Boundary

   https://doi.org/10.1515/agms-2022-0141  

   Gibara, Ryan; Shanmugalingam, Nageswari (January 2022, Analysis and Geometry in Metric Spaces)

   Abstract The sphericalization procedure converts a Euclidean space into a compact sphere. In this note we propose a variant of this procedure for locally compact, rectifiably path-connected, non-complete, unbounded metric spaces by using conformal deformations that depend only on the distance to the boundary of the metric space. This deformation is locally bi-Lipschitz to the original domain near its boundary, but transforms the space into a bounded domain. We will show that if the original metric space is a uniform domain with respect to its completion, then the transformed space is also a uniform domain. 

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