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1. Coexistence Phenomena in the Hénon Family

   https://doi.org/10.1007/s00574-023-00345-9  

   Benedicks, Michael ; Palmisano, Liviana ( July 2023 , Bulletin of the Brazilian Mathematical Society, New Series)

   We study the classical Hénon family$$f_{a,b}:(x,y)\mapsto (1-ax^2+y,bx)$$$f_{a,b}:(x,y)\mapsto (1-a{x}^2+y,bx)$,$$0$0<a<2$,$$0$0<b<1$, and prove that given an integer$$k\ge 1$$$k\ge 1$, there is a set of parameters$$E_k$$$E_k$of positive two-dimensional Lebesgue measure so that$$f_{a,b}$$$f_{a,b}$, for$$(a,b)\in E_k$$$(a,b)\in E_k$, has at leastkattractive periodic orbits and one strange attractor of the type studied in Benedicks and Carleson (Ann Math (2) 133(1):73–169, 1991). A corresponding statement also holds for the Hénon-like families of Mora and Viana (Acta Math 171:1–71, 1993), and we use the techniques of Mora and Viana (1993) to study homoclinic unfoldings also in the case of the original Hénon maps. The final main result of the paper is the existence, within the classical Hénon family, of a positive Lebesgue measure set of parameters whose corresponding maps have two coexisting strange attractors.

    

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2. Measurement of non-prompt $${{\textrm{D}}^{0}}$$-meson elliptic flow in Pb–Pb collisions at $$\sqrt{s_{\textrm{NN}}} = 5.02$$ TeV

   https://doi.org/10.1140/epjc/s10052-023-12259-3  

   Acharya, S ; Adamová, D ; Aglieri_Rinella, G ; Agnello, M ; Agrawal, N ; Ahammed, Z ; Ahmad, S ; Ahn, S U ; Ahuja, I ; Akindinov, A ; et al ( December 2023 , The European Physical Journal C)

   The elliptic flow$$(v_2)$$$\left(v_2\right)$of$${\textrm{D}}^{0}$$${\text{D}}^0$mesons from beauty-hadron decays (non-prompt$${\textrm{D}}^{0})$$${\text{D}}^0)$was measured in midcentral (30–50%) Pb–Pb collisions at a centre-of-mass energy per nucleon pair$$\sqrt{s_{\textrm{NN}}} = 5.02$$$\sqrt{s_{\text{NN}}}=5.02$ TeV with the ALICE detector at the LHC. The$${\textrm{D}}^{0}$$${\text{D}}^0$mesons were reconstructed at midrapidity$$(|y|<0.8)$$$\left(\right|y|<0.8)$from their hadronic decay$$\mathrm {D^0 \rightarrow K^-\uppi ^+}$$$D^0\to K^{-}{\pi }^{+}$, in the transverse momentum interval$$2< p_{\textrm{T}} < 12$$$2<p_{\text{T}}<12$ GeV/c. The result indicates a positive$$v_2$$$v_2$for non-prompt$${{\textrm{D}}^{0}}$$${\text{D}}^0$mesons with a significance of 2.7$$\sigma $$$\sigma$. The non-prompt$${{\textrm{D}}^{0}}$$${\text{D}}^0$-meson$$v_2$$$v_2$is lower than that of prompt non-strange D mesons with 3.2$$\sigma $$$\sigma$significance in$$2< p_\textrm{T} < 8~\textrm{GeV}/c$$$2<p_{\text{T}}<8\text{GeV}/c$, and compatible with the$$v_2$$$v_2$of beauty-decay electrons. Theoretical calculations of beauty-quark transport in a hydrodynamically expanding medium describe the measurement within uncertainties.

    

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3. Multipliers on bi-parameter Haar system Hardy spaces

   https://doi.org/10.1007/s00208-024-02887-9  

   Lechner, R. ; Motakis, P. ; Müller, P_F_X ; Schlumprecht, Th ( May 2024 , Mathematische Annalen)

   Let$$(h_I)$$$\left(h_I\right)$denote the standard Haar system on [0, 1], indexed by$$I\in \mathcal {D}$$$I\in D$, the set of dyadic intervals and$$h_I\otimes h_J$$$h_I\otimes h_J$denote the tensor product$$(s,t)\mapsto h_I(s) h_J(t)$$$(s,t)\mapsto h_I\left(s\right)h_J\left(t\right)$,$$I,J\in \mathcal {D}$$$I,J\in D$. We consider a class of two-parameter function spaces which are completions of the linear span$$\mathcal {V}(\delta ^2)$$$V\left({\delta }^2\right)$of$$h_I\otimes h_J$$$h_I\otimes h_J$,$$I,J\in \mathcal {D}$$$I,J\in D$. This class contains all the spaces of the formX(Y), whereXandYare either the Lebesgue spaces$$L^p[0,1]$$$L^p[0,1]$or the Hardy spaces$$H^p[0,1]$$$H^p[0,1]$,$$1\le p < \infty $$$1\le p<\infty$. We say that$$D:X(Y)\rightarrow X(Y)$$$D:X\left(Y\right)\to X\left(Y\right)$is a Haar multiplier if$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$$D(h_I\otimes h_J)=d_{I,J}{h}_I\otimes h_J$, where$$d_{I,J}\in \mathbb {R}$$$d_{I,J}\in R$, and ask which more elementary operators factor throughD. A decisive role is played by theCapon projection$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$$C:V\left({\delta }^2\right)\to V\left({\delta }^2\right)$given by$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$$C{h}_I\otimes h_J=h_I\otimes h_J$if$$|I|\le |J|$$$\left|I\right|\le \left|J\right|$, and$$\mathcal {C} h_I\otimes h_J = 0$$$C{h}_I\otimes h_J=0$if$$|I| > |J|$$$\left|I\right|>\left|J\right|$, as our main result highlights: Given any bounded Haar multiplier$$D:X(Y)\rightarrow X(Y)$$$D:X\left(Y\right)\to X\left(Y\right)$, there exist$$\lambda ,\mu \in \mathbb {R}$$$\lambda ,\mu \in R$such that$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$$$\begin{array}{c}\lambda C+\mu (\text{Id}-C)\text{approximately 1-projectionally factors through}D,\end{array}$i.e., for all$$\eta > 0$$$\eta >0$, there exist bounded operatorsA, Bso thatABis the identity operator$${{\,\textrm{Id}\,}}$$$\text{Id}$,$$\Vert A\Vert \cdot \Vert B\Vert = 1$$$\Vert A\Vert ·\Vert B\Vert =1$and$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta $$$\Vert \lambda C+\mu (\text{Id}-C)-ADB\Vert <\eta$. Additionally, if$$\mathcal {C}$$$C$is unbounded onX(Y), then$$\lambda = \mu $$$\lambda =\mu$and then$${{\,\textrm{Id}\,}}$$$\text{Id}$either factors throughDor$${{\,\textrm{Id}\,}}-D$$$\text{Id}-D$.

    

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4. Perfectly Packing a Square by Squares of Nearly Harmonic Sidelength

   https://doi.org/10.1007/s00454-023-00523-y  

   Tao, Terence ( July 2023 , Discrete & Computational Geometry)

   A well-known open problem of Meir and Moser asks if the squares of sidelength 1/nfor$$n\ge 2$$$n\ge 2$can be packed perfectly into a rectangle of area$$\sum _{n=2}^\infty n^{-2}=\pi ^2/6-1$$${\sum }_{n=2}^{\infty }{n}^{-2}={\pi }^2/6-1$. In this paper we show that for any$$1/2$1/2<t<1$, and any$$n_0$$$n_0$that is sufficiently large depending on t, the squares of sidelength$$n^{-t}$$$n^{-t}$for$$n\ge n_0$$$n\ge n_0$can be packed perfectly into a square of area$$\sum _{n=n_0}^\infty n^{-2t}$$${\sum }_{n=n_0}^{\infty }{n}^{-2t}$. This was previously known (if one packs a rectangle instead of a square) for$$1/2$1/2<t\le 2/3$(in which case one can take$$n_0=1$$$n_0=1$).

    

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5. Green function estimates on complements of low-dimensional uniformly rectifiable sets

   https://doi.org/10.1007/s00208-022-02379-8  

   David, Guy ; Feneuil, Joseph ; Mayboroda, Svitlana ( March 2022 , Mathematische Annalen)

   It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$$L_{\beta ,\gamma }=-\text{div}{D}^{d+1+\gamma -n}\nabla$associated to a domain$$\Omega \subset {\mathbb {R}}^n$$$\Omega \subset R^n$with a uniformly rectifiable boundary$$\Gamma $$$\Gamma$of dimension$$d < n-1$$$d<n-1$, the now usual distance to the boundary$$D = D_\beta $$$D=D_{\beta }$given by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$$D_{\beta }{\left(X\right)}^{-\beta }={\int }_{\Gamma }{|X-y|}^{-d-\beta }d\sigma \left(y\right)$for$$X \in \Omega $$$X\in \Omega$, where$$\beta >0$$$\beta >0$and$$\gamma \in (-1,1)$$$\gamma \in (-1,1)$. In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$$L_{\beta ,\gamma }$, with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$$D^{1-\gamma }$, in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$$|D\nabla (ln(\frac{G}{D^{1-\gamma }})){|}^2$satisfies a Carleson measure estimate on$$\Omega $$$\Omega$. We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).

    

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