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1. Universal minimal flows of extensions of and by compact groups

   https://doi.org/10.1017/etds.2022.52  

   BARTOŠOVÁ, DANA (August 2023, Ergodic Theory and Dynamical Systems)

   Abstract Every topological group G has, up to isomorphism, a unique minimal G -flow that maps onto every minimal G -flow, the universal minimal flow $M(G).$ We show that if G has a compact normal subgroup K that acts freely on $M(G)$ and there exists a uniformly continuous cross-section from $G/K$ to $G,$ then the phase space of $M(G)$ is homeomorphic to the product of the phase space of $M(G/K)$ with K . Moreover, if either the left and right uniformities on G coincide or G is isomorphic to a semidirect product $$G/K\ltimes K$$ , we also recover the action, in the latter case extending a result of Kechris and Sokić. As an application, we show that the phase space of $M(G)$ for any totally disconnected locally compact Polish group G with a normal open compact subgroup is homeomorphic to a finite set, the Cantor set $$2^{\mathbb {N}}$$ , $$M(\mathbb {Z})$$ , or $$M(\mathbb {Z})\times 2^{\mathbb {N}}.$$ 

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2. Non-stationary Itô–Kawada and ergodic theorems for random isometries

   https://doi.org/10.4171/GGD/856  

   Monakov, Grigorii V (January 2025, Groups, Geometry, and Dynamics)

   We consider a non-stationary sequence of independent random isometries of a compact metrizable space. Assuming that there are no proper closed subsets with deterministic image, we establish a weak-* convergence to the unique invariant under isometries measure, ergodic theorem and large deviation type estimate. We also show that all the results can be carried over to the case of a random walk on a compact metrizable group. In particular, we prove a non-stationary analog of classical Itô–Kawada theorem and give a new alternative proof for the stationary case. 

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3. On Polish groups admitting non-essentially countable actions

   https://doi.org/10.1017/etds.2020.133  

   KECHRIS, ALEXANDER S.; MALICKI, MACIEJ; PANAGIOTOPOULOS, ARISTOTELIS; ZIELINSKI, JOSEPH (January 2022, Ergodic Theory and Dynamical Systems)

   Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable. 

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4. Uniqueness and stability of Ricci flow through singularities

   https://doi.org/DOI: https://dx.doi.org/10.4310/ACTA.2022.v228.n1.a1  

   Bamler, Richard; Kleiner, Bruce (July 2022, Acta mathematica)

   We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3‑manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman’s conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of 3‑manifolds—in particular to the generalized Smale conjecture—which will appear in a subsequent paper. 

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5. Langevin dynamic for the 2D Yang–Mills measure

   https://doi.org/10.1007/s10240-022-00132-0  

   Chandra, Ajay; Chevyrev, Ilya; Hairer, Martin; Shen, Hao (June 2022, Publications mathématiques de l'IHÉS)

   Abstract We define a natural state space and Markov process associated to the stochastic Yang–Mills heat flow in two dimensions. To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric. To construct the Markov process we show that the stochastic Yang–Mills heat flow takes values in our space of connections and use the “DeTurck trick” of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations. Our main tool for solving for the Yang–Mills heat flow is the theory of regularity structures and along the way we also develop a “basis-free” framework for applying the theory of regularity structures in the context of vector-valued noise – this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest. 

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