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1. Structure of partially hyperbolic Hénon maps

   https://doi.org/10.4171/JEMS/1074  

   Lyubich, Mikhail; Peters, Hans (January 2021, Journal of the European Mathematical Society)

   We describe the structure of “substantially dissipative” complex Hénon maps admitting a dominated splitting on the Julia set. We prove that the Fatou set consists of only finitely many components, each either attracting or parabolic periodic. In particular, there are no wandering components and no rotation domains. 

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2. Finiteness Properties of Affine Difference Algebraic Groups

   https://doi.org/10.1093/imrn/rnaa177  

   Wibmer, Michael (July 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the matrix entries can indeed be defined by finitely many such equations. As an application, we show that the difference ideal of all difference algebraic relations among the solutions of a linear differential equation is finitely generated. 

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3. COH, SRT 2 2 , and multiple functionals

   https://doi.org/10.3233/COM-190261  

   Dzhafarov, Damir D.; Patey, Ludovic (April 2021, Computability)

   null (Ed.)

   We prove the following result: there is a family R = ⟨ R 0 , R 1 , … ⟩ of subsets of ω such that for every stable coloring c : [ ω ] 2 → k hyperarithmetical in R and every finite collection of Turing functionals, there is an infinite homogeneous set H for c such that none of the finitely many functionals map R ⊕ H to an infinite cohesive set for R. This provides a partial answer to a question in computable combinatorics, whether COH is omnisciently computably reducible to SRT 2 2 . 

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4. Stability and the inverse gravimetry problem with minimal data

   https://doi.org/10.1515/jiip-2020-0115  

   Isakov, Victor; Titi, Aseel (November 2020, Journal of Inverse and Ill-posed Problems)

   Abstract The inverse problem in gravimetry is to find a domain 𝐷 inside the reference domain Ω from boundary measurements of gravitational force outside Ω.We found that about five parameters of the unknown 𝐷 can be stably determined given data noise in practical situations.An ellipse is uniquely determined by five parameters.We prove uniqueness and stability of recovering an ellipse for the inverse problem from minimal amount of data which are the gravitational force at three boundary points.In the proofs, we derive and use simple systems of linear and nonlinear algebraic equations for natural parameters of an ellipse.To illustrate the technique, we use these equations in numerical examples with various location of measurements points on ∂ ⁡ Ω \partial\Omega .Similarly, a rectangular 𝐷 is considered.We consider the problem in the plane as a model for the three-dimensional problem due to simplicity. 

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5. Poincaré constant on manifolds with ends

   https://doi.org/10.1112/plms.12522  

   Grigor'yan, Alexander; Ishiwata, Satoshi; Saloff‐Coste, Laurent (April 2023, Proceedings of the London Mathematical Society)

   Abstract We obtain optimal estimates of the Poincaré constant of central balls on manifolds with finitely many ends. Surprisingly enough, the Poincaré constant is determined by thesecondlargest end. The proof is based on the argument by Kusuoka–Stroock where the heat kernel estimates on the central balls play an essential role. For this purpose, we extend earlier heat kernel estimates obtained by the authors to a larger class of parabolic manifolds with ends. 

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