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1. Zero entropy actions of amenable groups are not dominant

   https://doi.org/10.1017/etds.2023.17  

   LOTT, ADAM (March 2023, Ergodic Theory and Dynamical Systems)

   Abstract A probability measure-preserving action of a discrete amenable groupGis said to bedominantif it is isomorphic to a generic extension of itself. Recently, it was shown that for$$G = \mathbb {Z}$$, an action is dominant if and only if it has positive entropy and that for anyG, positive entropy implies dominance. In this paper, we show that the converse also holds for anyG, that is, that zero entropy implies non-dominance. 

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2. Equivalence of codes for countable sets of reals

   https://doi.org/10.4153/S0008439520000661  

   Chan, William (September 2021, Canadian Mathematical Bulletin)

   null (Ed.)

   Abstract A set $$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$$ is universal for countable subsets of $${\mathbb {R}}$$ if and only if for all $$x \in {\mathbb {R}}$$ , the section $$U_x = \{y \in {\mathbb {R}} : U(x,y)\}$$ is countable and for all countable sets $$A \subseteq {\mathbb {R}}$$ , there is an $$x \in {\mathbb {R}}$$ so that $$U_x = A$$ . Define the equivalence relation $$E_U$$ on $${\mathbb {R}}$$ by $$x_0 \ E_U \ x_1$$ if and only if $$U_{x_0} = U_{x_1}$$ , which is the equivalence of codes for countable sets of reals according to U . The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $$E_{U^*}$$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $$E_U$$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $$E_U$$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $$\mathbf {\Sigma }_3^1$$ -generic absoluteness, then for all $$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$$ that are $$\mathbf {\Sigma }_1^1$$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $$E_U$$ . 

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3. Entropy, products, and bounded orbit equivalence

   https://doi.org/10.1017/etds.2021.154  

   KERR, DAVID; LI, HANFENG (March 2023, Ergodic Theory and Dynamical Systems)

   Abstract We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, then the actions have the same sofic topological entropy. This fact is then used to show that if two free uniquely ergodic and entropy regular probability-measure-preserving actions of such groups are boundedly orbit equivalent then the actions have the same sofic measure entropy. Our arguments are based on a relativization of property SC to sofic approximations and yield more general entropy inequalities. 

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4. Ancient solutions and translators of Lagrangian mean curvature flow

   https://doi.org/10.1007/s10240-023-00143-5  

   Lotay, Jason D; Schulze, Felix; Székelyhidi, Gábor (January 2024, Publications mathématiques de l'IHÉS)

   Abstract Suppose that ℳ is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in$$\mathbf {C} ^{n}$$ $C^n$. We show that if ℳ has a blow-down given by the static union of two Lagrangian subspaces with distinct Lagrangian angles that intersect along a line, then ℳ is a translator. In particular in$$\mathbf {C} ^{2}$$ $C^2$, all almost calibrated, exact, ancient solutions of Lagrangian mean curvature flow with entropy less than 3 are special Lagrangian, a union of planes, or translators. 

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5. Computability of topological pressure on compact shift spaces beyond finite type*

   https://doi.org/10.1088/1361-6544/ac7702  

   Burr, Michael; Das, Suddhasattwa; Wolf, Christian; Yang, Yun (June 2022, Nonlinearity)

   We investigate the computability (in the sense of computable analysis) of the topological pressure P_top(ϕ) on compact shift spaces X for continuous potentials ϕ:X→R. This question has recently been studied for subshifts of finite type (SFTs) and their factors (sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalised gap shifts, and particular beta-shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further prove that the generalised pressure function (X,ϕ) ↦P_top(X,ϕ|_X) is not computable for a large set of shift spaces X and potentials ϕ . In particular, the entropy map X↦h_top(X) is computable at a shift spaceXif and only if X has zero topological entropy. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability. 

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