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1. C*-algebras of a Cantor system with finitely many minimal subsets: structures, K-theories, and the index map

   https://doi.org/10.1017/etds.2020.12  

   BEZUGLYI, SERGEY; NIU, ZHUANG; SUN, WEI (May 2021, Ergodic Theory and Dynamical Systems)

   We study homeomorphisms of a Cantor set with$$k$$($$k<+\infty$$) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and certain of their orbit-cut sub-C*-algebras. In the case where$$k\geq 2$$, the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank 0 if in addition$$(X,\unicode[STIX]{x1D70E})$$is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli–Vershik–Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of the Bratteli–Vershik–Kakutani model) must have at least$$k$$vertices at each level, and the image of the index map must consist of infinitesimals. 

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2. On cohesive powers of linear orders

   https://doi.org/10.1017/jsl.2023.14  

   Dimitrov, Rumen; Harizanov, Valentina; Morozov, Andrey; Shafer, Paul; Soskova, Alexandra A; Vatev, Stefan V (September 2023, The Journal of Symbolic Logic)

   Abstract Cohesive powersof computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let$$\omega $$,$$\zeta $$, and$$\eta $$denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of$$\omega $$. If$$\mathcal {L}$$is a computable copy of$$\omega $$that is computably isomorphic to the usual presentation of$$\omega $$, then every cohesive power of$$\mathcal {L}$$has order-type$$\omega + \zeta \eta $$. However, there are computable copies of$$\omega $$, necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to$$\omega + \zeta \eta $$. For example, we show that there is a computable copy of$$\omega $$with a cohesive power of order-type$$\omega + \eta $$. Our most general result is that if$$X \subseteq \mathbb {N} \setminus \{0\}$$is a Boolean combination of$$\Sigma _2$$sets, thought of as a set of finite order-types, then there is a computable copy of$$\omega $$with a cohesive power of order-type$$\omega + \boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$$, where$$\boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$$denotes the shuffle of the order-types inXand the order-type$$\omega + \zeta \eta + \omega ^*$$. Furthermore, ifXis finite and non-empty, then there is a computable copy of$$\omega $$with a cohesive power of order-type$$\omega + \boldsymbol {\sigma }(X)$$. 

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3. Stable laws for random dynamical systems

   https://doi.org/10.1017/etds.2024.5  

   AIMINO, ROMAIN; NICOL, MATTHEW; TÖRÖK, ANDREW (February 2024, Ergodic Theory and Dynamical Systems)

   Abstract In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval$$[0,1]$$in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure$$\nu $$. We consider a class of non-square-integrable observables$$\phi $$, mostly of form$$\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$$, where$$x_0$$is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index$$\alpha \in (0,2)$$. The two types of maps we concatenate are a class of piecewise$$C^2$$expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and$$\alpha $$, we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by$$\nu $$. This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization. 

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4. Modular curves and Néron models of generalized Jacobians

   https://doi.org/10.1112/S0010437X23007662  

   Jordan, Bruce W; Ribet, Kenneth A; Scholl, Anthony J (May 2024, Compositio Mathematica)

   Let$$X$$be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring$$R$$, and$$\mathfrak {m}$$a modulus on$$X$$, given by a closed subscheme of$$X$$which is geometrically reduced. The generalized Jacobian$$J_\mathfrak {m}$$of$$X$$with respect to$$\mathfrak {m}$$is then an extension of the Jacobian of$$X$$by a torus. We describe its Néron model, together with the character and component groups of the special fibre, in terms of a regular model of$$X$$over$$R$$. This generalizes Raynaud's well-known description for the usual Jacobian. We also give some computations for generalized Jacobians of modular curves$$X_0(N)$$with moduli supported on the cusps. 

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5. Higher uniformity of arithmetic functions in short intervals I. All intervals

   https://doi.org/10.1017/fmp.2023.28  

   Matomäki, Kaisa; Shao, Xuancheng; Tao, Terence; Teräväinen, Joni (January 2023, Forum of Mathematics, Pi)

   Abstract We study higher uniformity properties of the Möbius function$$\mu $$, the von Mangoldt function$$\Lambda $$, and the divisor functions$$d_k$$on short intervals$$(X,X+H]$$with$$X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$$for a fixed constant$$0 \leq \theta < 1$$and any$$\varepsilon>0$$. More precisely, letting$$\Lambda ^\sharp $$and$$d_k^\sharp $$be suitable approximants of$$\Lambda $$and$$d_k$$and$$\mu ^\sharp = 0$$, we show for instance that, for any nilsequence$$F(g(n)\Gamma )$$, we have$$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when$$\theta = 5/8$$and$$f \in \{\Lambda , \mu , d_k\}$$or$$\theta = 1/3$$and$$f = d_2$$. As a consequence, we show that the short interval Gowers norms$$\|f-f^\sharp \|_{U^s(X,X+H]}$$are also asymptotically small for any fixedsfor these choices of$$f,\theta $$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in$$L^2$$. Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type$$II$$sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type$$I_2$$sums. 

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