We show that geometric disorder leads to some purely singular continuous spectrum generically. The main input is a result of Simon known as the Wonderland theorem in [17, Section 2]. Here we provide an alternative approach and actually a slight strengthening by showing that various sets of measures defined by regularity properties are generic in the set of all measures on a locally compact metric space. As a byproduct, we obtain the fact that a generic measure on euclidean space is singular continuous

We study the dynamics of a bimeromorphic map $X\to X$, where $X$ is a compact complex Kähler surface. Under a natural geometric hypothesis, we construct an invariant probability measure, which is mixing, hyperbolic, and of maximal entropy. The proof relies heavily on the theory of laminar currents and is new even in the case of polynomial automorphisms of ${\mathbb{C}}^2$. This extends recent results by E. Bedford and J. Diller

The Eisenstein-Picard modular group $\mathrm{PU}(2,1;\mathbb{Z}[\omega ])$ is defined to be the subgroup of $\mathrm{PU}(2,1)$ whose entries lie in the ring $\mathbb{Z}[\omega ]$, where $\omega$ is a cube root of unity. This group acts isometrically and properly discontinuously on $H_{\mathbb{C}}^2$, that is, on the unit ball in ${\mathbb{C}}^2$ with the Bergman metric. We construct a fundamental domain for the action of $\mathrm{PU}(2,1;\mathbb{Z}[\omega ])$ on $H_{\mathbb{C}}^2$, which is a 4-simplex with one ideal vertex. As a consequence, we elicit a presentation of the group (see Theorem 5.9). This seems to be the simplest fundamental domain for a finite covolume subgroup of $\mathrm{PU}(2,1)$

In this article, some asymptotic formulas are proved for the harmonic mollified second moment of a family of Rankin-Selberg $L$-functions. One of the main new inputs is a substantial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average. A first consequence is a new subconvexity bound for Rankin-Selberg $L$-functions in the level aspect. Moreover, infinitely many Rankin-Selberg $L$-functions having at most eight nontrivial real zeros are produced, and some new nontrivial estimates for the analytic rank of the family studied are obtained

We show that the set of faithful representations of a closed orientable hyperbolic surface group is dense in both irreducible components of the ${\mathrm{PSL}}_2(\mathbb{K})$ representation variety, where $\mathbb{K}=\mathbb{C}$ or $\mathbb{R}$, answering a question of W. M. Goldman. We also prove the existence of faithful representations into $\mathrm{PU}(2,1)$ with certain nonintegral Toledo invariants.

Suppose that $G$ and $H$ are connected reductive groups over a number field $F$ and that an $L$-homomorphism $\rho :{}^{L}G\to {}^{L}H$ is given. The Langlands functoriality conjecture predicts the existence of a map from the automorphic representations of $G(\mathbb{A})$ to those of $H(\mathbb{A})$. If the adelic points of the algebraic groups $G$, $H$ are replaced by their metaplectic covers, one may hope to specify an analogue of the $L$-group (depending on the cover), and then one may hope to construct an analogous correspondence. In this article, we construct such a correspondence for the double cover of the split special orthogonal groups, raising the genuine automorphic representations of ${\widetilde{\mathrm{SO}}}_{2k}(\mathbb{A})$ to those of ${\widetilde{\mathrm{SO}}}_{2k+1}(\mathbb{A})$. To do so, we use as integral kernel the theta representation on odd orthogonal groups constructed by the authors in a previous article [3]. In contrast to the classical theta correspondence, this representation is not minimal in the sense of corresponding to a minimal coadjoint orbit, but it does enjoy a smallness property in the sense that most conjugacy classes of Fourier coefficients vanish