Article
Mathematics
Qinbo Chen, Danijela Damjanovic
Summary: In this paper, we study local rigidity for isometric toral extensions of partially hyperbolic Zk (k ≥ 2) actions on the torus. We prove a C infinity local rigidity result for such actions, under the assumption that smooth perturbations of the actions satisfy the intersection property. We also provide a local rigidity result within a class of volume preserving actions. Our method mainly relies on a generalization of the Kolmogorov-Arnold-Moser iterative scheme.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics, Applied
Zeya Mi, Biao You, Yuntao Zang
Summary: In this paper, we prove that a partially hyperbolic attractor for a C-1 vector field with two dimensional center supports an SRB measure. Additionally, we show that in the case of a C-2 vector field, if the center bundle satisfies the sectionally expanding condition with respect to Gibbs u-states, then the attractor can only support finitely many SRB/physical measures.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Todd Fisher, Boris Hasselblatt
Summary: This paper establishes the C-1-density of stable accessibility in partially hyperbolic flows, as well as in the categories of volume-preserving, symplectic, and contact partially hyperbolic flows. The results include the C-1-density of topological transitivity, triviality of the centralizer, and the K-property of the natural volume.
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Applied
Alina Luchko, Igor Parasyuk
Summary: This study investigates an autonomous system with an invariant manifold M, discussing exponential stability, asymptotic phases, and geometric structures of approaching orbits. The exponential stability of the invariant manifold is addressed using Lyapunov functions, while the existence of asymptotic phases is shown for M with a partially hyperbolic structure. Moreover, it is demonstrated that a neighborhood of M has an invariant foliation structure, with each leaf corresponding to motions sharing a common asymptotic phase.
ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Ionut Chifan, Sayan Das, Bin Sun
Summary: By combining geometric methods in group theory with soft von Neumann algebraic techniques, we demonstrate that the von Neumann algebra L(Γ) of any icc, acylindrically hyperbolic group Γ satisfies the ISR property. This result has applications to various classes of groups such as hyperbolic groups, mapping class groups, and outer automorphisms of free groups. In addition, we obtain similar results for factors associated with groups that have nontrivial (quasi)cohomology valued into natural representations. The paper concludes by stating the positive answer to an open question posed by Amrutam and Jiang.©2023 Elsevier Inc. All rights reserved.
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Mathematics, Applied
Martin Mion-Mouton
Summary: This paper classifies and proves the three-dimensional partially hyperbolic diffeomorphisms with smooth distributions, highlighting the crucial role of the rigid geometric structure induced by the invariant distributions.
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2022)
Article
Mathematics
Yannick Guedes Bonthonneau, Thibault Lefeuvre
Summary: This article aims to extend a recent result on the local rigidity of the marked length spectrum from compact negatively-curved Riemannian manifolds to manifolds with hyperbolic cusps. We deal with the nonlinear version of the problem and prove that such manifolds are locally rigid for nonlinear perturbations of the metric that slightly decrease at infinity. Our proof relies on the linear theory and careful analytic study of the generalized X-ray transform operator.
JOURNAL DE L ECOLE POLYTECHNIQUE-MATHEMATIQUES
(2023)
Article
Mathematics, Applied
Zeya Mi
Summary: This paper studies physical measures for a class of partially hyperbolic flows with mostly contracting center. It proves that under certain conditions, these flows have finitely many physical measures, and their basins can cover almost all points.
DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL
(2021)
Article
Physics, Multidisciplinary
Elias Zafiris, Albrecht von Mueller
Summary: The quantum modular variables are encoded in terms of one-parameter unitary groups, leading to a re-evaluation of the Heisenberg group and its integral condition on the discrete structure. The structural transition from non-commutativity to its integral Abelian shadow is mediated by the discrete Heisenberg group, elucidating the role of modular variables in quantum mechanics and explaining the nature of quantum interference phenomena underlying the geometric phase concept.
QUANTUM STUDIES-MATHEMATICS AND FOUNDATIONS
(2022)
