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1. Ergodic theorem for nonstationary random walks on compact abelian groups

   https://doi.org/10.1090/proc/16848  

   Monakov, Grigorii (July 2024, Proceedings of the American Mathematical Society)

   We consider a nonstationary random walk on a compact metrizable abelian group. Under a classical strict aperiodicity assumption we establish a weak-* convergence to the Haar measure, Ergodic Theorem and Large Deviation Type Estimate. 

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2. Topology of Gleason Parts in Maximal Ideal Spaces with no Analytic Discs

   https://doi.org/10.4153/S0008414X19000567  

   Izzo, Alexander J.; Papathanasiou, Dimitris (February 2021, Canadian Journal of Mathematics)

   null (Ed.)

   Abstract We strengthen, in various directions, the theorem of Garnett that every $$\unicode[STIX]{x1D70E}$$ -compact, completely regular space $$X$$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $$X$$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $$X$$ of a Euclidean space, there is a compact set $$K$$ in some $$\mathbb{C}^{N}$$ so that $$\widehat{K}\backslash K$$ contains a Gleason part homeomorphic to  $$X$$ , and $$\widehat{K}$$ contains no analytic discs. 

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3. Non-Stationary Version of Ergodic Theorem for Random Dynamical Systems

   https://doi.org/10.17323/1609-4514-2023-23-4-515-532  

   Gorodetski, Anton; Kleptsyn, Victor (October 2023, Moscow Mathematical Journal)

   Ilyashenko, Yu; Tsfasman, M; Gusein-Zade, S (Ed.)

   We prove a version of pointwise ergodic theorem for non- stationary random dynamical systems. Also, we discuss two specificc examples where the result is applicable: non-stationary iterated function systems and non-stationary random matrix products. 

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4. Maximally highly proximal flows

   https://doi.org/10.1017/etds.2020.40  

   ZUCKER, ANDY (May 2020, Ergodic Theory and Dynamical Systems)

   For G a Polish group, we consider G-flows which either contain a comeager orbit or have all orbits meager. We single out a class of flows, the maximally highly proximal (MHP) flows, for which this analysis is particularly nice. In the former case, we provide a complete structure theorem for flows containing comeager orbits, generalizing theorems of Melleray, Nguyen Van Thé, and Tsankov and of Ben Yaacov, Melleray, and Tsankov. In the latter, we show that any minimal MHP flow with all orbits meager has a metrizable factor with all orbits meager, thus ‘reflecting’ complicated dynamical behavior to metrizable flows. We then apply this to obtain a structure theorem for Polish groups whose universal minimal flow is distal. 

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5. Non-stationary Anderson localization

   https://doi.org/10.1090/cams/45  

   Gorodetski, Anton; Kleptsyn, Victor (January 2025, Communications of the American Mathematical Society)

   We consider discrete Schrödinger operators on ${\mathrm{\ell <\#comment/>}}^2\left(\mathbb{Z}\right)$with bounded random but not necessarily identically distributed values of the potential. We prove spectral localization (with exponentially decaying eigenfunctions) as well as dynamical localization for this model. An important ingredient of the proof is a non-stationary version of the parametric Furstenberg Theorem on random matrix products, which is also of independent interest. 

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