Denjoy minimal sets and Birkhoff periodic orbits for non-exact monotone twist maps

A non-exact monotone twist map ${\overline{\phi }}_F$ is a composition of an exact monotone twist map $\overline{\phi }$ with a generating function H and a vertical translation $V_F$ with $V_F((x,y))=(x,y?F)$. We show in this paper that for each $\omega \in \mathbb{R}$, there exists a critical value $F_d(\omega )\ge 0$ depending on H and ω such that for $0\le F\le F_d(\omega )$, the non-exact twist map ${\overline{\phi }}_F$ has an invariant Denjoy minimal set with irrational rotation number ω lying on a Lipschitz graph, or Birkhoff $(p,q)$-periodic orbits for rational $\omega =p/q$. Like the Aubry–Mather theory, we also construct heteroclinic orbits connecting Birkhoff periodic orbits, and show that quasi-periodic orbits in these Denjoy minimal sets can be approximated by periodic orbits. In particular, we demonstrate that at the critical value $F=F_d(\omega )$, the Denjoy minimal set is not uniformly hyperbolic and can be approximated by smooth curves.     

A remark on the orbit structure of the complexification of a semisimple symmetric space

We consider the action of a real semisimple Lie group G on the complexification $G_{\mathbf{C}}/H_{\mathbf{C}}$ of a semisimple symmetric space $G/H$ and we present a refinement of Matsuki?s results (Matsuki, 1997 [1]) in this case. We exhibit a finite set of points in $G_{\mathbf{C}}/H_{\mathbf{C}}$, sitting on closed G-orbits of locally minimal dimension, whose slice representation determines the G-orbit structure of $G_{\mathbf{C}}/H_{\mathbf{C}}$. Every such point $\overline{p}$ lies on a compact torus and occurs at specific values of the restricted roots of the symmetric pair $(\mathfrak{g},\mathfrak{h})$. The slice representation at $\overline{p}$ is equivalent to the isotropy representation of a real reductive symmetric space, namely $Z_G(p^4)/G_{\overline{p}}$. In principle, this gives the possibility to explicitly parametrize all G-orbits in $G_{\mathbf{C}}/H_{\mathbf{C}}$.     

Extended Picard complexes for algebraic groups and homogeneous spaces

For a smooth geometrically integral algebraic variety X over a field k of characteristic 0, we define the extended Picard complex $\mathrm{UPic}(\overline{X})$. It is a complex of length 2 which combines the Picard group $\mathrm{Pic}(\overline{X})$ and the group $U(\overline{X}):=\overline{k}{[\overline{X}]}^{\times }/{\overline{k}}^{\times }$, where $\overline{k}$ is a fixed algebraic closure of k and $\overline{X}=X{\times }_k\overline{k}$. For a connected linear k-group G we compute the complex $\mathrm{UPic}(\overline{G})$ (up to a quasi-isomorphism) in terms of the algebraic fundamental group ${\pi }_1(\overline{G})$. We obtain similar results for a homogeneous space X of a connected k-group G. To cite this article: M. Borovoi, J. van Hamel, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On overpartition pairs into odd parts modulo powers of 2

In 2012, Lin (Electron. J. Combin. 19(2) (2012) #P17) investigated the 2 and 3-divisibility properties for ${\overline{pp}}_o\left(n\right)$, the number of overpartition pairs into odd parts. Using modular forms, he proved that for a fixed positive integer $k$, ${\overline{pp}}_o\left(n\right)$ is almost always divisible by $2^k$. In this paper, we prove several congruences for ${\overline{pp}}_o\left(n\right)$ modulo higher powers of 2 in an elementary way.     

Frequentistic approximations to Bayesian prevision of exchangeable random elements

Posterior and predictive distributions for m future trials, given the first n elements of an infinite exchangeable sequence ${\tilde{\xi }}_1,{\tilde{\xi }}_2,\dots$, are considered in a nonparametric Bayesian setting. The former distribution is compared to the unit mass at the empirical distribution ${\tilde{\mathfrak{e}}}_n:=\frac{1}{n}{\sum }_{i=1}^n{\delta }_{{\tilde{\xi }}_i}$ of the n past observations, while the latter is compared to the m-fold product ${\tilde{\mathfrak{e}}}_n^m$. Comparisons are made by means of distinguished probability distances inducing topologies that are equivalent to (or finer than) the topology of weak convergence of probability measures. After stating almost sure convergence to zero of these distances as n goes to infinity, the paper focuses on the analysis of the rate of approach to zero, so providing a quantitative evaluation of the approximation of posterior and predictive distributions through their frequentistic counterparts ${\delta }_{{\tilde{\mathfrak{e}}}_n}$ and ${\tilde{\mathfrak{e}}}_n^m$, respectively. Characteristic features of the present work, with respect to more common literature on Bayesian consistency, are: first, comparisons are made between entities which depend on the n past observation only; second, the approximations are studied under the actual (exchangeable) law of the ${\tilde{\xi }}_n$'s, and not under hypothetical product laws ${\mathfrak{p}}_0^{\infty }$, as ${\mathfrak{p}}_0$ varies among the admissible determinations of a random probability measure.