We prove a central limit theorem for a class of Hölder continuous cocycles with an application to stricly convex and irreducible rational representations of hyperbolic groups, introduced by Sambarino [Quantitative properties of convexe representations. Comment. Math. Helv 89 (2014), 443-488].
We propose a notion of critical set for two-dimensional surface diffeomorphisms as an intrinsically defined object designed to play a role analogous to that of critical points in one-dimensional dynamics.
If two homogeneous IFSs satisfying the OSC with opposite common contraction factors share the same attractor on the real line, we show that this attractor is symmetric. This answers a question of Feng and Wang [Adv. Math. 222 (2009), 1964-1981].
We construct an explicit family of finite-area, infinite-genus translation surfaces whose vertical translation flow is strongly mixing. This provides a positive answer to a question posed by Lindsey and Treviño~\cite{LT}
Recently, gene-based association studies have shown that integrating genome-wide association studies (GWAS) with expression quantitative trait locus (eQTL) data can boost statistical power and that the genetic liability of traits can be captured by polygenic risk scores (PRSs). In this paper, we propose a new gene-based statistical method that leverages gene-expression measurements and new PRSs to identify genes that are associated with phenotypes of interest. We used a generalized linear model to associate phenotypes with gene expression and PRSs and used a score-test statistic to test the association between phenotypes and genes. Our simulation studies show that the newly developed method has correct type I error rates and can boost statistical power compared with other methods that use either gene expression or PRS in association tests. A real data analysis figure based on UK Biobank data for asthma shows that the proposed method is applicable to GWAS.
We show the coexistence of chaotic behaviors (positive metric entropy) and elliptic behaviors (intregrable KAM island) among analytic, symplectic diffeomorphism of any closed surface. In particilar this solves a problem by F. Przytycki (1982).
On an elliptic billard, we study a complex reflexion law introduced by A. Glutsyuk in order to state that the locus of the circumcenters of triangular orbits is an ellipse.
For any polynomial diffeomorphism $f$ of ${\Bbb C}^2$ with positive entropy, neither the Julia set of $f$ nor of its inverse $f^{-1}$ is semi-analytic.
We provide a proof of Pisot conjecture, a classification problem in Ergodic Theory on recurrent sequences generated by irreducible Pisot substitutions.
We give a geometrical demonstration to the existence of holomorphic first integrals for certain kind of vector fields in $\mathbb{C}^2$ and $\mathbb{C}^3$.
Modifying Hall's idea in "A C^{\infty} Denjoy counterexample" we construct an example of homeomorphism of the circle which is a Denjoy counterexample (i.e. it is not conjugated to a rotation) and which is a C^{\infty}-diffeomorphism everywhere except in a flat half-critical point.
We introduce a double staircase construction $T$ and show that the weak closure of $\{T^n\}$ is $\{\int, 2^{-m}T^n+(1-2^{-m})\int \ : m\in\N,\ n\in \Z\}.$
A map $f$ on a compact metric space is expansive if and only if $f^n$ is expansive. We study the exponential rate of decay of the expansive constant of $f^n$. A major result is that this rate times box dimension bounds topological entropy.
We prove the decoration theorem for the Mandelbrot set (and Multibrot sets) which says that when a "little Mandelbrot set" is removed from the Mandelbrot set, then most of the resulting connected components have small diameters.