Article
Mathematics
Sebastian Hurtado, Jinxin Xue
Summary: We provide a complete classification for polycyclic abelian-by-cyclic group actions on T-2 under mild assumptions, considering both topological and smooth conjugacy. Additionally, we prove a Tits alternative-type theorem for some groups of diffeomorphisms of T-2.
GEOMETRY & TOPOLOGY
(2021)
Article
Mathematics, Applied
Qiao Liu
Summary: This paper studies a local rigidity property, where any smooth perturbations close enough to an affine action can be smoothly conjugate to the affine action with constant time change.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2021)
Article
Mathematics, Applied
Homin Lee
Summary: In this article, it is proven that for a C(infinity) volume-preserving action α on a closed n-manifold M, by a lattice Γ in SL(n, R), there exists an element γ ∈ Γ such that a(γ) has a dominated splitting. It is also shown that M is diffeomorphic to the n-dimensional Torus and α is smoothly conjugate to an affine action. Similar theorems are proved for actions on 2n-manifolds by lattices in Sp(2n, R) with n >= 2, and SO(n, n) with n >= 5.
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Applied
Adam Kanigowski, Philipp Kunde, Kurt Vinhage, Daren Wei
Summary: We study slow entropy invariants for abelian unipotent actions U on any finite volume homogeneous space G/Gamma. For every such action, we show that the topological slow entropy can be computed directly from the dimension of a special decomposition of Lie(G) induced by Lie(U). Moreover, we prove that the metric slow entropy of the action coincides with its topological slow entropy. As a corollary, we generalize the rank one results from [14] to higher rank abelian actions.
JOURNAL OF MODERN DYNAMICS
(2022)
Article
Mathematics, Applied
Andrey Gogolev, Federico Rodriguez Hertz
Summary: We introduce the technique of matching functions in the context of Anosov flows. We show that simple periodic cycle functionals can be used as matching functions for conjugate Anosov flows. These simple periodic cycle functionals are C1 regular for conservative codimension one Anosov flows, which improves the regularity of the conjugacy. Specifically, we prove that a continuous conjugacy must be a C1 diffeomorphism for an open and dense set of codimension one conservative Anosov flows.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics, Applied
Minsung Kim
Summary: The main result of this paper is the construction of finitely additive measures for higher rank abelian actions on Heisenberg nilmanifolds. Under a full measure set of Diophantine conditions for the generators of the action, Bufetov functionals are constructed on (2g + 1)-dimensional Heisenberg manifolds. The paper proves that the deviation of the ergodic integral of higher rank actions can be described by the asymptotic of Bufetov functionals for a sufficiently smooth function. As a corollary, the distribution of normalized ergodic integrals with variance 1 converges to a non-degenerate compactly supported measure on the real line along certain subsequences.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2022)
Article
Mathematics
David E. Evans, Ulrich Pennig
Summary: This paper develops an equivariant Dixmier-Douady theory for locally trivial bundles of C-*-algebras, studying the fibrewise T-action and its associated automorphism group and cohomology theory. The paper computes the group of isomorphism classes for equivariant bundles with respect to the fibrewise tensor product and compares it to the equivariant Brauer group for trivial actions.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics
Tiago J. Fonseca
Summary: We describe higher dimensional generalizations of Ramanujan's classical differential relations satisfied by the Eisenstein series E2, E4, E6. These higher Ramanujan equations are given geometrically in terms of vector fields living on certain moduli stacks classifying abelian schemes equipped with suitable frames of their first de Rham cohomology. Using Mumford's theory of degenerating families of abelian varieties, we construct remarkable solutions of these differential equations generalizing (E2, E4, E6), which are also shown to be defined over Z. This geometric framework taking account of integrality issues is mainly motivated by questions in Transcendental Number Theory regarding an extension of Nesterenko's celebrated theorem on the algebraic independence of values of Eisenstein series.
MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Physics, Particles & Fields
Spyros Konitopoulos
Summary: A systematic method is presented for the construction and classification of algebras of gauge transformations for arbitrary high rank tensor gauge fields. By adjusting the values of the initial coefficients and imposing restrictions on the gauge functions, one can recover previously analyzed algebras and construct new ones. The presentation of a new algebra for tensor gauge transformations is the central result of this article.
Article
Mathematics
Jonathan Bowden, Sebastian Hensel, Kathryn Mann, Emmanuel Militon, Richard Webb
Summary: Building on previous work [7], this study explores the action of the homeomorphism group on the fine curve graph of a surface. Unlike the classical curve graph for mapping class groups, the action of homeomorphisms on this graph exhibits a richer dynamics: parabolic isometries in addition to elliptic and hyperbolic ones, and the realization of all positive reals as asymptotic translation lengths. When the surface is a torus, the dynamics of the homeomorphism action on the fine curve graph is related to the dynamics on the torus using the classical theory of rotation sets.
ADVANCES IN MATHEMATICS
(2022)
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
Andrey Gogolev, Boris Kalinin, Victoria Sadovskaya
Summary: In this paper, we study the perturbations of a partially hyperbolic toral automorphism L that is diagonalizable over C and has a dense center foliation. We establish the existence of a smooth leaf conjugacy to L for a small perturbation with a smooth center foliation. We also show that if a small perturbation of an ergodic irreducible L has a smooth center foliation and is bi-Holder conjugate to L, then the conjugacy is smooth. As a corollary, we prove that any bi-Holder conjugacy of a symplectic perturbation of such an L must be smooth. For a totally irreducible L with a two-dimensional center, we establish several equivalent conditions on the perturbation that ensure smooth conjugacy to L.
MATHEMATISCHE ANNALEN
(2023)
Article
Automation & Control Systems
Fernando Pereira Micena, Rafael de la Llave
Summary: This study investigates rigidity results by analyzing regular points in the context of Oseledec's Theorem, and examines the possibility of Anosov diffeomorphisms having all Lyapunov exponents defined everywhere. The condition of having Lyapunov exponents defined everywhere implies local rigidity of an Anosov automorphism of the torus T-d.
JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS
(2021)
Article
Mathematics
Fanjun Meng, Mihnea Popa
Summary: We provide proofs for the additivity result of log Kodaira dimension on algebraic fiber spaces over abelian varieties, the superadditivity result for fiber spaces over varieties of maximal Albanese dimension, as well as the subadditivity result for log pairs.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
Article
Mathematics, Applied
Fernando Micena
Summary: We prove smooth conjugacy between non-necessarily special Anosov endomorphisms in the conservative case. In addition, we show that a strongly special C degrees degrees-Anosov endomorphism of T2 and its linearization are smoothly conjugated because they share the same periodic data. Assuming that every point of a strongly special C degrees degrees-Anosov endomorphism of T2 is regular (in the sense of Oseledec's Theorem), we again obtain smooth conjugacy with its linearization. We also obtain results on the local rigidity of linear Anosov endomorphisms of d-torus, where d >= 3, under periodic data assumption. The study of differential equations defined on invariant leaves is important in rigidity problems like those discussed in this paper.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2023)
Article
Mathematics
Zhiren Wang
INVENTIONES MATHEMATICAE
(2017)
Article
Mathematics
Zhiren Wang
JOURNAL D ANALYSE MATHEMATIQUE
(2018)
Article
Mathematics, Applied
Matthew Litman, Zhiren Wang
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2019)
Article
Mathematics
Wen Huang, Zhiren Wang, Xiangdong Ye
ADVANCES IN MATHEMATICS
(2019)
Article
Mathematics
Xiaoguang He, Zhiren Wang
Summary: The decay of the averaged short interval correlation on a nilmanifold G/Gamma was studied, with a uniform bound observed as the intervals grow infinite in length. This bound applies to all elements in G, x in G/Gamma, and the continuous function F used in the analysis.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Amir Algom, Federico Rodriguez Hertz, Zhiren Wang
Summary: This paper provides conditions on the derivative cocycle to ensure that every self conformal measure is supported on points x that are absolutely normal, while also extending state-of-the-art results. When Φ is self-similar, the set of absolutely normal numbers has full Hausdorff dimension in its attractor, unless Φ has a certain explicit structure. These conditions on the derivative cocycle also imply that every self conformal measure is a Rajchman measure.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics
Amir Algom, Federico Rodriguez Hertz, Zhiren Wang
Summary: We prove that the Fourier transform of a self-conformal measure on the real axis decays to 0 at a logarithmic rate, unless certain conditions are met. Our key technical result is an effective version of a local limit theorem for cocycles with moderate deviations, which is of independent interest.
JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES
(2022)
Article
Mathematics
Aaron Brown, Federico Hertz, Zhiren Wang
Summary: This article investigates the lattice actions of higher-rank semisimple Lie groups on manifolds. We introduce two numbers, r(G) and m(G), which are associated with the roots system of the Lie algebra of a Lie group G. We prove that if the dimension of the manifold satisfies certain conditions, the actions preserve or relatively preserve measures.
ANNALS OF MATHEMATICS
(2022)
Article
Mathematics, Applied
Amir Algom, Zhiren Wang
Summary: Sarnak's Mobius disjointness conjecture investigates the mathematical properties of zero entropy dynamical systems. We constructed examples to verify the conjecture and found that o(1) can approach zero at arbitrarily slow speeds.
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2022)
Proceedings Paper
Computer Science, Artificial Intelligence
Yi Wang, Zhiren Wang
Summary: This research demonstrates that under certain assumptions, stochastic gradient descent has three stages including descent, diffusion and tunneling, reaching a temporary equilibrium state, explaining its relationship with the fast equilibrium conjecture.
INTERNATIONAL CONFERENCE ON MACHINE LEARNING, VOL 162
(2022)
Article
Mathematics, Applied
Federico Rodriguez Hertz, Zhiren Wang
Summary: The paper proves that the Hausdorff codimension of the particular set Z(epsilon),D in a finite volume homogeneous space of a semisimple Lie group G is at least c epsilon, where c depends only on G, G(0) and Gamma.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2021)
Article
Mathematics, Applied
Wen Huang, Zhiren Wang, Guohua Zhang
JOURNAL OF MODERN DYNAMICS
(2019)
Article
Mathematics, Applied
Xiao-Yuan Wang, Lei Shi, Zhi-Ren Wang
JOURNAL OF COMPLEX ANALYSIS
(2018)
Article
Mathematics
Aaron Brown, Federico Rodriguez Hertz, Zhiren Wang
ANNALS OF MATHEMATICS
(2017)
