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  1. Abstract

    Every Thurston map$f\colon S^2\rightarrow S^2$on a$2$-sphere$S^2$induces a pull-back operation on Jordan curves$\alpha \subset S^2\smallsetminus {P_f}$, where${P_f}$is the postcritical set off. Here the isotopy class$[f^{-1}(\alpha )]$(relative to${P_f}$) only depends on the isotopy class$[\alpha ]$. We study this operation for Thurston maps with four postcritical points. In this case, a Thurston obstruction for the mapfcan be seen as a fixed point of the pull-back operation. We show that if a Thurston mapfwith a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can ‘blow up’ suitable arcs in the underlying$2$-sphere and construct a new Thurston map$\widehat f$for which this obstruction is eliminated. We prove that no other obstruction arises and so$\widehat f$is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points. We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem.

     
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    Free, publicly-accessible full text available September 1, 2025
  2. Abstract. We study existence and uniqueness of Green functions for the Cheeger Q- Laplacian in metric measure spaces that are Ahlfors Q-regular and support a Q-Poincar ́e inequality with Q > 1. We prove uniqueness of Green functions both in the case of relatively compact domains, and in the global (unbounded) case. We also prove existence of global Green functions in unbounded spaces, complementing the existing results in relatively compact domains proved recently in [BBL20]. 
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    Free, publicly-accessible full text available April 19, 2025
  3. 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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  4. Freiberg, U ; null (Ed.)
  5. null (Ed.)
    Abstract We show that if an expanding Thurston map is the quotient of a torus endomorphism, then it has a parabolic orbifold and is a Lattès-type map. 
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