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1. Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, II

   https://doi.org/10.1515/crelle-2017-0024  

   Brock, Jeffrey; Leininger, Christopher; Modami, Babak; Rafi, Kasra (January 2020, Journal für die reine und angewandte Mathematik (Crelles Journal))

   null (Ed.)

   Abstract Given a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of which projectively converges to one of the ergodic measures on the ending lamination. The conditions are sufficiently robust, allowing us to construct sequences on a closed surface of genus g for which the space of measures has the maximal dimension {3g-3} , for example. We also study the limit sets in the Thurston boundary of Teichmüller geodesic rays defined by quadratic differentials whose vertical foliations are obtained from the constructions mentioned above. We prove that such examples exist for which the limit is a cycle in the 1-skeleton of the simplex of projective classes of measures visiting every vertex. 

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2. Endperiodic automorphisms of surfaces and foliations

   https://doi.org/10.1017/etds.2019.56  

   CANTWELL, JOHN; CONLON, LAWRENCE; FENLEY, SERGIO R. (January 2021, Ergodic Theory and Dynamical Systems)

   We extend the unpublished work of Handel and Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel–Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic laminations, show that the geodesic laminations satisfy the axioms, and prove that pseudo-geodesic laminations satisfying our axioms are ambiently isotopic to the geodesic laminations. The axiomatic approach allows us to show that the given endperiodic automorphism is isotopic to a smooth endperiodic automorphism preserving smooth laminations ambiently isotopic to the original ones. Using the axioms, we also prove the ‘transfer theorem’ for foliations of 3-manifolds, namely that, if two depth-one foliations $${\mathcal{F}}$$ and $${\mathcal{F}}^{\prime }$$ are transverse to a common one-dimensional foliation $${\mathcal{L}}$$ whose monodromy on the non-compact leaves of $${\mathcal{F}}$$ exhibits the nice dynamics of Handel–Miller theory, then $${\mathcal{L}}$$ also induces monodromy on the non-compact leaves of $${\mathcal{F}}^{\prime }$$ exhibiting the same nice dynamics. Our theory also applies to surfaces with infinitely many ends. 

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3. An Extension of the Kaliszewski Cone to Non-polyhedral Pointed Cones in Infinite-Dimensional Spaces

   https://doi.org/https://doi.org/10.1007/s10957-018-01468-6  

   Huerga, Lidia; Jadamba, Baasansuren; Sama, Miguel (May 2019, Journal of optimization theory and applications)

   In this paper, we propose an extension of the family of constructible dilating cones given by Kaliszewski (Quantitative Pareto analysis by cone separation technique, Kluwer Academic Publishers, Boston, 1994) from polyhedral pointed cones in finite-dimensional spaces to a general family of closed, convex, and pointed cones in infinite-dimensional spaces, which in particular covers all separable Banach spaces. We provide an explicit construction of the new family of dilating cones, focusing on sequence spaces and spaces of integrable functions equipped with their natural ordering cones. Finally, using the new dilating cones, we develop a conical regularization scheme for linearly constrained least-squares optimization problems. We present a numerical example to illustrate the efficacy of the proposed framework. 

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4. Limit Sets of Weil–Petersson Geodesics

   https://doi.org/10.1093/imrn/rny002  

   Brock, Jeffrey; Leininger, Christopher; Modami, Babak; Rafi, Kasra (February 2018, International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper we prove that the limit set of any Weil–Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichmüller space. On the other hand, we construct examples of Weil–Petersson geodesics with minimal non-uniquely ergodic ending laminations and limit set a circle in the Thurston compactification. 

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5. Dynamical generation of parameter laminations

   https://doi.org/10.1090/conm/744  

   Blokh, A; Oversteegen, L; Timorin, V. (January 2020, Contemporary mathematics)

   Abstract. Local similarity between the Mandelbrot set and quadratic Julia sets manifests itself in a variety of ways. We discuss a combinatorial one, in the language of geodesic laminations. More precisely, we compare quadratic invariant laminations representing Julia sets with the so-called Quadratic Minor Lamination (QML) representing a locally connected model of the Mandelbrot set. Similarly to the construction of an invariant lamination by pullbacks of certain leaves, we describe how QML can be generated by properly understood pullbacks of certain minors. In particular, we show that the minors of all non-renormalizable quadratic laminations can be obtained by taking limits of “pullbacks” of minors from the main cardioid 

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