Geometric and Functional Analysis最新文献

We study semiclassical measures for Laplacian eigenfunctions on compact complex hyperbolic quotients. Geodesic flows on these quotients are a model case of hyperbolic dynamical systems with different expansion/contraction rates in different directions. We show that the support of any semiclassical measure is either equal to the entire cosphere bundle or contains the cosphere bundle of a compact immersed totally geodesic complex submanifold.

The proof uses the one-dimensional fractal uncertainty principle of Bourgain–Dyatlov (Ann. Math. (2) 187(3):825–867, 2018) along the fast expanding/contracting directions, in a way similar to the work of Dyatlov–Jézéquel (Ann. Henri Poincaré, 2023) in the toy model of quantum cat maps, together with a description of the closures of fast unstable/stable trajectories relying on Ratner theory.

Abstract

Despite increasing data on the properties of the origins of replication, the molecular mechanisms underlying the origin recognition complex (ORC) positioning in the genome are still poorly understood. It has been suggested that the key factors determining the positioning of ORC in the genome are DNA-binding proteins that form various DNA regulatory elements, including insulators, promoters, and enhancers, thereby linking the replication program to different levels of transcriptional regulation. Previously, we demonstrated that the Su(Hw) protein is the first example of such a protein. Subsequent studies identified a number of other DNA-binding proteins, including CG10543, which may be responsible for the formation of the corresponding regulatory elements and the recruitment of transcriptional and replication complexes to their binding sites. It has been shown that the Drosophila CG10543 protein interacts with the deubiquitinating (DUB) module of the SAGA complex. The binding sites of the CG10543 protein are predominantly located in the promoter regions of active genes and colocalize with the SAGA and dSWI/SNF chromatin modification and remodeling complexes, as well as with the ORC replication complex. To investigate the role of the CG10543 protein in transcriptional regulation, an RNA-Seq experiment was conducted in Drosophila S2 cells under normal conditions and upon RNA interference with the CG10543 protein. It was shown that the CG10543 protein affects the transcription of 469 genes, with a significant portion of these genes (23%) being ecdysone-dependent genes. Ecdysone is the main steroid hormone in Drosophila, is responsible for Drosophila metamorphosis, and has a significant effect on the expression of many genes during development. We demonstrated that CG10543 sites colocalize with the CBP protein and the histone mark H3K27Ac, which are characteristic of active regulatory elements. The CG10543 protein also colocalizes with the CP190 protein, suggesting a potential mechanism of transcriptional regulation through the formation of long-range interactions between regulatory elements.

We establish an explicit global spectral decomposition of shifted convolution sums and the second moment of automorphic (L)-functions for Maaß forms with explicit integral transforms as well as explicit inversion formulae over every local field.

We prove that one hundred percent of the closed geodesic periods of a Hecke–Maaß cusp form for the modular group are non-vanishing when ordered by length. We present applications to the non-vanishing of central values of Rankin–Selberg (L)-functions. Similar results for holomorphic forms for general Fuchsian groups of finite covolume with a cusp are also obtained, as well as results towards normal distribution. Our new key ingredient is to relate the distributions of closed geodesic periods and vertical line integrals via graph theory.

We prove foundational results about the set of homomorphisms from a finitely generated group to the collection of all fundamental groups of compact 3–manifolds and answer questions of Agol–Liu (J. Am. Math. Soc. 25(1):151–187, 2012) and Reid–Wang–Zhou (Acta Math. Sin. Engl. Ser. 18(1):157–172, 2002).

By constructing a non-empty domain of discontinuity in a suitable homogeneous space, we prove that every torsion-free projective Anosov subgroup is the monodromy group of a locally homogeneous contact Axiom A dynamical system with a unique basic hyperbolic set on which the flow is conjugate to the refraction flow of Sambarino. Under the assumption of irreducibility, we utilize the work of Stoyanov to establish spectral estimates for the associated complex Ruelle transfer operators, and by way of corollary: exponential mixing, exponentially decaying error term in the prime orbit theorem, and a spectral gap for the Ruelle zeta function. With no irreducibility assumption, results of Dyatlov-Guillarmou imply the global meromorphic continuation of zeta functions with smooth weights, as well as the existence of a discrete spectrum of Ruelle-Pollicott resonances and (co)-resonant states. We apply our results to space-like geodesic flows for the convex cocompact pseudo-Riemannian manifolds of Danciger-Guéritaud-Kassel, and the Benoist-Hilbert geodesic flow for strictly convex real projective manifolds.

Let (M,g₀) be a closed oriented hyperbolic manifold of dimension at least 3. By the volume entropy inequality of G. Besson, G. Courtois and S. Gallot, for any Riemannian metric g on M with same volume as g₀, its volume entropy h(g) satisfies h(g)≥n−1 with equality only when g is isometric to g₀. We show that the hyperbolic metric g₀ is stable in the following sense: if g_i is a sequence of Riemaniann metrics on M of same volume as g₀ and if h(g_i) converges to n−1, then there are smooth subsets Z_i⊂M such that both (operatorname{Vol}(Z_{i},g_{i})) and (operatorname{Area}(partial Z_{i},g_{i})) tend to 0, and (M∖Z_i,g_i) converges to (M,g₀) in the measured Gromov-Hausdorff topology. The proof relies on showing that any spherical Plateau solution for M is intrinsically isomorphic to ((M,frac{(n-1)^{2}}{4n} g_{0})).

The main purpose of this paper is to propose an ergodic theoretic approach to the study of entire holomorphic curves. Brody curves are one-Lipschitz holomorphic maps from the complex plane to the complex projective space. They admit a natural group action, and “random Brody curves” in the title refers to invariant probability measures for it. We study their geometric and dynamical properties. Given an invariant probability measure μ on the space of Brody curves, our first main theorem claims that its rate distortion dimension is bounded by the integral of a “geometric potential” over μ. This result is analogous to the Ruelle inequality of smooth ergodic theory. Our second main theorem claims that there exists a rich variety of invariant probability measures attaining equality in this “Ruelle inequality for Brody curves”. The main tools of the proofs are the deformation theory of Brody curves and the variational principle for mean dimension with potential. This approach is motivated by the theory of thermodynamic formalism for Axiom A diffeomorphisms.

Let G be a countable group that is the fundamental group of a graph of groups with finite edge groups and vertex groups satisfying the strong Atiyah conjecture over (K subseteq mathbb{C}) a field closed under complex conjugation. Assume that the orders of finite subgroups of G are bounded above. We show that G satisfies the strong Atiyah conjecture over K. In particular, this implies that the strong Atiyah conjecture is closed under free products. Moreover, we prove that the ∗-regular closure of K[G] in (mathcal{U}(G)), (mathcal{R}_{K[G]}), is a universal localization of the graph of rings associated to the graph of groups, where the rings are the corresponding ∗-regular closures. As a result, we obtain that the algebraic and center-valued Atiyah conjecture over K are also closed under the graph of groups construction as long as the edge groups are finite. We also infer some consequences on the structure of the K₀ and K₁-groups of (mathcal{R}_{K[G]}). The techniques developed enable us to prove that K[G] fulfills the strong, algebraic and center-valued Atiyah conjectures, and that (mathcal{R}_{K[G]}) is the universal localization of K[G] over the set of all matrices that become invertible in (mathcal{U}(G)), provided that G belongs to a certain class of groups (mathcal{T}_{mathcal{VLI}}), which contains in particular virtually-{locally indicable} groups that are the fundamental group of a graph of virtually free groups.