Article
Mathematics, Applied
A. Avila, Marcelo Viana, A. Wilkinson
Summary: In this study, we explore new connections between the dynamics of conservative partially hyperbolic systems and the geometric measure-theoretic properties of their invariant foliations. We investigate two main classes of volume-preserving diffeomorphisms: fibered partially hyperbolic diffeomorphisms and center-fixing partially hyperbolic systems. We establish that, under certain assumptions, the disintegration of the volume measure along the center foliation is either atomic or Lebesgue. In addition, for three-dimensional cases, we show that the center foliation is smooth and the diffeomorphism is smoothly conjugate to an explicit rigid model.
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2022)
Article
Mathematics
Svetlana Jitomirskaya, Sasa Kocic
Summary: The study focuses on Schrodinger operators with ergodic potentials defined over circle map dynamics, especially over circle diffeomorphisms. For analytic circle diffeomorphisms with rotation numbers satisfying certain arithmetic conditions, an extension of Avila's global theory is discussed, along with a sharp Gordon-type theorem on the absence of eigenvalues for potentials with repetitions. The results show that, coupled with dynamical analysis, Schrodinger operators associated with certain circle diffeomorphisms have purely continuous spectrum for certain continuous potentials.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2022)
Article
Mathematics
Zeya Mi, Yongluo Cao
Summary: This study establishes a topological structure called skeleton for partially hyperbolic diffeomorphisms. It investigates the continuity of physical measures with respect to dynamics under C-1-topology by studying perturbations on the skeletons.
MATHEMATISCHE ZEITSCHRIFT
(2021)
Article
Mathematics, Applied
Shaobo Gan, Yi Shi, Disheng Xu, Jinhua Zhang
Summary: This paper investigates the centralizer of a partially hyperbolic diffeomorphism on T-3, showing that the centralizer is virtually trivial or the diffeomorphism is smoothly conjugate to its linear part.
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2021)
Article
Mathematics, Applied
Meg Doucette
Summary: This paper proves that under restrictions on the fiber, any fibered partially hyperbolic system over a nilmanifold can be leaf conjugate to a smooth model that is isometric on the fibers and descends to a hyperbolic nilmanifold automorphism on the base. One key ingredient is a result generalizing a result of Hiraide: an Anosov homeomorphism of a nilmanifold is topologically conjugate to a hyperbolic nilmanifold automorphism.
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Applied
Alexey V. Ivanov
Summary: In this paper, we study a skew product F-A = (s(?), A) over an irrational rotation s(?) (x) = x + ? of a circle T-1. It is assumed that the transformation A : T-1 → SL(2, R), a C-1-map, has the form A(x) = R(?(x))Z(?(x)), where R(?) is a rotation in R-2 through the angle ? and Z(?) = diag{?, ?(-1)} is a diagonal matrix. By considering the condition ?(x) = ?(0) > 1 with a sufficiently large constant ?(0) and the property that cos ?(x) possesses only simple zeroes, we investigate the hyperbolic properties of the cocycle generated by F-A. Using the critical set method, we show that under certain additional requirements on the derivative of the function ?, the secondary collisions compensate for the weakening of hyperbolicity caused by primary collisions, resulting in a uniformly hyperbolic cocycle generated by F-A, in contrast to the case where secondary collisions can be partially eliminated.
REGULAR & CHAOTIC DYNAMICS
(2023)
Article
Mathematics, Applied
Weisheng Wu
Summary: This paper investigates certain partially hyperbolic diffeomorphisms with a centre of arbitrary dimension and demonstrates the continuity properties of the topological entropy under C (1) perturbations. The systems studied exhibit subexponential growth in the center direction and uniform exponential growth along the unstable foliation. The findings apply to PHDs that are Lyapunov stable in the center direction, as well as another important class of systems with subexponential growth in the center direction, for which a technique is developed to achieve uniform distribution of unstable manifolds using the exponential mixing property of the systems.
Article
Mathematics, Applied
Laura Caravenna, Gianluca Crippa
Summary: The article proves a Lipschitz extension lemma that simultaneously preserves Lipschitz continuity for two nonequivalent distances, addressing the issue of high integrability assumption in the DiPerna-Lions theory.
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Physics, Mathematical
Guoqiao You, Changfeng Xue, Shaozhong Deng
Summary: In this paper, effective Eulerian algorithms are introduced for the computation of the forward finite time Lyapunov exponent (FTLE) of smooth flow fields. The proposed algorithms improve the accuracy of the numerical solution and are computationally efficient. Numerical examples demonstrate their effectiveness.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Weisheng Wu, Fei Liu, Fang Wang
Summary: In this paper, the ergodicity of geodesic flows on surfaces without focal points is studied. Under the assumption that the set of points on the surface with negative curvature has at most finitely many connected components, it is proved that the geodesic flow on the unit tangent bundle is ergodic with respect to the Liouville measure.
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2023)
Article
Mathematics
Rafael O. Ruggiero
Summary: This paper proves that under certain conditions, the geodesic flow is Anosov. Combining recent results, it is concluded that if the geodesic flow is structurally stable from a certain viewpoint, then it is an Anosov flow.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics, Interdisciplinary Applications
Kei Inoue, Kazuki Tani
Summary: This paper introduces a quantification method for chaos in traffic flow models. The extended entropic chaos degree can directly compute the chaos level of time series with lower computational complexity. Through empirical research, it is demonstrated that the extended chaos degree can be used to quantify chaos in traffic flow models.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Astronomy & Astrophysics
Shobhit Giri, Hemwati Nandan, Lokesh Kumar Joshi, Sunil D. Maharaj
Summary: This study investigates the existence and stability of timelike and null circular orbits for a (2 + 1)-dimensional charged BTZ black hole in Einstein-nonlinear Maxwell gravity with a negative cosmological constant. The analysis of orbit stability explores the potential chaos in geodesic motion within a special case of conformally invariant Maxwell spacetime. Calculations of proper time Lyapunov exponent and coordinate time Lyapunov exponent are utilized to assess the stability of circular orbits.
MODERN PHYSICS LETTERS A
(2021)
Article
Mathematics, Applied
Yu-Keung Ng, Guoqiao You, Shingyu Leung
Summary: We propose an efficient approach to estimate the finite-time Lyapunov exponent (FTLE) by developing a sparse subsampling method to detect relevant flow measurements for velocity reconstruction, instead of considering all available velocity measurements. This work has two main contributions: extending a previous algorithm to reconstruct a flow field under an impermeable condition, and proposing a L1 optimization framework for solving the corresponding under-determined system of reconstructing the global velocity field, which leads to a sparse reconstruction algorithm. Synthetic and real-life numerical examples will be provided to demonstrate the effectiveness of the proposed algorithm.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Physics, Mathematical
Lingrui Ge, Jiangong You, Xin Zhao
Summary: This paper proves the Holder continuity of the integrated density of states for a class of quasi-periodic long-range operators with large trigonometric polynomial potentials and Diophantine frequencies. Additionally, the Holder exponent is given in terms of the cardinality of the level sets of the potentials, improving upon previous results in the perturbative regime.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2022)
Article
Statistics & Probability
Elais C. Malheiro, Marcelo Viana
STOCHASTICS AND DYNAMICS
(2015)
Article
Physics, Mathematical
Dmitry Dolgopyat, Marcelo Viana, Jiagang Yang
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2016)
Article
Mathematics, Applied
Carlos Bocker-Neto, Marcelo Viana
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2017)
Article
Mathematics, Applied
Vanessa Ramos, Marcelo Viana
Article
Mathematics, Applied
Marcelo Viana, Jiagang Yang
ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
(2013)
Article
Mathematics, Applied
Marcelo Viana
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2020)
Article
Mathematics
Artur Avila, Marcelo Viana
INVENTIONES MATHEMATICAE
(2010)
Article
Mathematics, Applied
Gang Liao, Marcelo Viana, Jiagang Yang
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
(2013)
Editorial Material
Mathematics
Marcelo Viana
BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY
(2019)
Article
Mathematics
Marcelo Viana, Jiagang Yang
ISRAEL JOURNAL OF MATHEMATICS
(2019)
Article
Mathematics, Applied
Mauricio Poletti, Marcelo Viana
Article
Mathematics
El Hadji Yaya Tall, Marcelo Viana
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2020)
Article
Mathematics
Artur Avila, Jimmy Santamaria, Marcelo Viana, Amie Wilkinson
Article
Mathematics
Artur Avila, Jimmy Santamaria, Marcelo Viana
