Katok’s entropy formula is an important formula in entropy theory.It plays significant roles in large deviation theories,multifractal analysis,quantitative recurrence and so on.This paper is devoted to establishing K...Katok’s entropy formula is an important formula in entropy theory.It plays significant roles in large deviation theories,multifractal analysis,quantitative recurrence and so on.This paper is devoted to establishing Katok’s entropy formula of unstable metric entropy which is the entropy caused by the unstable part of partially hyperbolic systems.We also construct a similar formula which can be used to study the quantitative recurrence in the unstable manifold for partially hyperbolic diffeomorphisms.展开更多
Let （M, ω） be a closed symplectic 2n-dimensional manifold. Donaldson in his paper showed that there exist 2m-dimensional symplectie submanifolds （V^2m,ω） of （M,ω）, 1 ≤m ≤ n - 1, with （m - 1）-equivalent in...Let （M, ω） be a closed symplectic 2n-dimensional manifold. Donaldson in his paper showed that there exist 2m-dimensional symplectie submanifolds （V^2m,ω） of （M,ω）, 1 ≤m ≤ n - 1, with （m - 1）-equivalent inclusions. On the basis of this fact we obtain isomorphic relations between kernel of Lefschetz map of M and kernels of Lefschetz maps of Donaldson submanifolds V^2m, 2 ≤ m ≤ n - 1. Then, using this relation, we show that the flux group of M is discrete if the action of π1 （M） on π2（M） is trivial and there exists a retraction r ： M→ V, where V is a 4-dimensional Donaldson submanifold. And, in the symplectically aspherical case, we investigate the flux groups of the manifolds.展开更多
Let T：X → X be an Axiom A diffeomorphism,m the Gibbs state for a Hlder continuous function ɡ. Assume that f：X → R^d is a Hlder continuous function with ∫_X^（fdm） = 0.If the components of f are cohomologously i...Let T：X → X be an Axiom A diffeomorphism,m the Gibbs state for a Hlder continuous function ɡ. Assume that f：X → R^d is a Hlder continuous function with ∫_X^（fdm） = 0.If the components of f are cohomologously independent, then there exists a positive definite symmetric matrix σ~2：=σ~2 （f ） such that S^fn √ n converges in distribution with respect to m to a Gaussian random variable with expectation 0 and covariance matrix σ~2 . Moreover, there exists a real number A 〉 0 such that, for any integer n ≥ 1,Π（ m＊（ 1√ nS f n ）,N （0,σ~2 ） ≤A√n, where m＊（1√ n S^fn）denotes the distribution of 1√ n S^fn with respect to m, and Π is the Prokhorov metric.展开更多
Viewing gravitational energy-momentum PG<sup style='margin-left:-7px;'>μ as equal by observation, but different in essence from inertial energy-momentum PI<sup style='margin-left:-7px;'>μ...Viewing gravitational energy-momentum PG<sup style='margin-left:-7px;'>μ as equal by observation, but different in essence from inertial energy-momentum PI<sup style='margin-left:-7px;'>μ naturally leads to the gauge theory of volume-preserving diffeomorphisms of a four-dimensional inner space. To analyse scattering in this theory, the gauge field is coupled to two Dirac fields with different masses. Based on a generalized LSZ reduction formula the S-matrix element for scattering of two Dirac particles in the gravitational limit and the corresponding scattering cross-section are calculated to leading order in perturbation theory. Taking the non-relativistic limit for one of the initial particles in the rest frame of the other the Rutherford-like cross-section of a non-relativistic particle scattering off an infinitely heavy scatterer calculated quantum mechanically in Newtonian gravity is recovered. This provides a non-trivial test of the gauge field theory of volume-preserving diffeomorphisms as a quantum theory of gravity.展开更多
In this paper we first survey the results on the embedding flow problem of dif-feomorphisms in higher dimensional spaces. Next we present some new results on the characterization of semi-unipotent diffeomorphisms in R...In this paper we first survey the results on the embedding flow problem of dif-feomorphisms in higher dimensional spaces. Next we present some new results on the characterization of semi-unipotent diffeomorphisms in R3, which have a formal embedding flows.展开更多
Viewing gravitational energy-momentum as equal by observation, but different in essence from inertial energymomentum naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner Minkowski space...Viewing gravitational energy-momentum as equal by observation, but different in essence from inertial energymomentum naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner Minkowski space which can describe gravitation at the classical level. This theory is quantized in the path integral formalism starting with a non-covariant Hamiltonian formulation with unconstrained canonical field variables and a manifestly positive Hamiltonian. The relevant path integral measure and weight are then brought into a Lorentz- and gauge-covariant form allowing to express correlation functions—applying the De Witt-Faddeev-Popov approach—in any meaningful gauge. Next the Feynman rules are developed and the quantum effective action at one loop in a background field approach is renormalized which results in an asymptotically free theory without presence of other fields and in a theory without asymptotic freedom including the Standard Model (SM) fields. Finally the BRST apparatus is developed as preparation for the renormalizability proof to all orders and a sketch of this proof is given.展开更多
Let ACD（M, SL（d,R）） denote the pairs （f, A） so that f∈ A C Diff^1（M） is a C^1-Anosov transitive diffeomorphisms and A is an SL（d,R） cocycle dominated with respect to f. We prove that open and densely in ACD...Let ACD（M, SL（d,R）） denote the pairs （f, A） so that f∈ A C Diff^1（M） is a C^1-Anosov transitive diffeomorphisms and A is an SL（d,R） cocycle dominated with respect to f. We prove that open and densely in ACD（M, SL（d,R））, in appropriate topologies, the pair （f,A） has simple spectrum with respect to the unique maximal entropy measure μf. Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in AUtLeb（M） × LP（M, SL（d, R））.展开更多
A suffcient condition for a set of calibrated submanifolds to be area-minimizing with multiplicities,also call weighted area-minimizing under diffeomorphisms (WAMD) is stated.We construct some WAMD submanifolds by ass...A suffcient condition for a set of calibrated submanifolds to be area-minimizing with multiplicities,also call weighted area-minimizing under diffeomorphisms (WAMD) is stated.We construct some WAMD submanifolds by assembling pieces of special Lagrangian (SL) normal bundles including the one of three surfaces meeting at an angle of 120° along soap-film-like singularities.We also mention a symmetry property of SL submanifolds and Bjrling type problem for SL normal bundles.展开更多
We prove that the restrictions of the conjugacy between two Anosov diffeomorphisms of the twotorus to the stable and unstable manifolds are quasisymmetric homeomorphisms.
We discuss the twistor quantization problem for the classical system (V d ,A d ), represented by the phase space V d , identified with the Sobolev space H 0 1/2 (S 1,? d ) of half-differentiable vector functions on th...We discuss the twistor quantization problem for the classical system (V d ,A d ), represented by the phase space V d , identified with the Sobolev space H 0 1/2 (S 1,? d ) of half-differentiable vector functions on the circle, and the algebra of observables A d , identified with the semi-direct product of the Heisenberg algebra of V d and the algebra Vect(S 1) of tangent vector fields on the circle.展开更多
Inertial and gravitational mass or energy momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner four-dime...Inertial and gravitational mass or energy momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The gauge-fixed action and the path integral measure occurring in the generating functional for the quantum Green functions of the theory are shown to obey a BRST-type symmetry. The related Zinn-Justin-type equation restricting the corresponding quantum effective action is established. This equation limits the infinite parts of the quantum effective action to have the same form as the gauge-fixed Lagrangian of the theory proving its spacetime renormalizability. The inner space integrals occurring in the quantum effective action which are divergent due to the gauge group’s infinite volume are shown to be regularizable in a way consistent with the symmetries of the theory demonstrating as a byproduct that viable quantum gauge field theories are not limited to finite-dimensional compact gauge groups as is commonly assumed.展开更多
In this paper,we will use the embedding flows in[1],[2]to give a complete descrip- tion of the smooth centralizers and iterate radicals of all C^r(r≥2)Morse-Smale diffeomorphisms of the circle S^1.As a result,we prov...In this paper,we will use the embedding flows in[1],[2]to give a complete descrip- tion of the smooth centralizers and iterate radicals of all C^r(r≥2)Morse-Smale diffeomorphisms of the circle S^1.As a result,we prove that every centralizer is a solvable subgroup of Diff^r(S^1).展开更多
基金supported by NNSF of China(12101446,11401581,11971236)The second author was supported by NNSF of China(11401581)+1 种基金the third author was supported by NNSF of China(11671208,11431012)At the end,we would like to express our gratitude to Tianyuan Mathematical Center in Southwest China,Sichuan University and Southwest Jiaotong University for their support and hospitality.
文摘Katok’s entropy formula is an important formula in entropy theory.It plays significant roles in large deviation theories,multifractal analysis,quantitative recurrence and so on.This paper is devoted to establishing Katok’s entropy formula of unstable metric entropy which is the entropy caused by the unstable part of partially hyperbolic systems.We also construct a similar formula which can be used to study the quantitative recurrence in the unstable manifold for partially hyperbolic diffeomorphisms.
文摘Let （M, ω） be a closed symplectic 2n-dimensional manifold. Donaldson in his paper showed that there exist 2m-dimensional symplectie submanifolds （V^2m,ω） of （M,ω）, 1 ≤m ≤ n - 1, with （m - 1）-equivalent inclusions. On the basis of this fact we obtain isomorphic relations between kernel of Lefschetz map of M and kernels of Lefschetz maps of Donaldson submanifolds V^2m, 2 ≤ m ≤ n - 1. Then, using this relation, we show that the flux group of M is discrete if the action of π1 （M） on π2（M） is trivial and there exists a retraction r ： M→ V, where V is a 4-dimensional Donaldson submanifold. And, in the symplectically aspherical case, we investigate the flux groups of the manifolds.
基金supported by the National Natural Science Foundation of China(10571174)the Scientific Research Foundation of Ministry of Education for Returned Overseas Chinese ScholarsScientific Research Foundation of Ministry of Human Resources and Social Security for Returned Overseas Chinese Scholars
文摘Let T：X → X be an Axiom A diffeomorphism,m the Gibbs state for a Hlder continuous function ɡ. Assume that f：X → R^d is a Hlder continuous function with ∫_X^（fdm） = 0.If the components of f are cohomologously independent, then there exists a positive definite symmetric matrix σ~2：=σ~2 （f ） such that S^fn √ n converges in distribution with respect to m to a Gaussian random variable with expectation 0 and covariance matrix σ~2 . Moreover, there exists a real number A 〉 0 such that, for any integer n ≥ 1,Π（ m＊（ 1√ nS f n ）,N （0,σ~2 ） ≤A√n, where m＊（1√ n S^fn）denotes the distribution of 1√ n S^fn with respect to m, and Π is the Prokhorov metric.
文摘Viewing gravitational energy-momentum PG<sup style='margin-left:-7px;'>μ as equal by observation, but different in essence from inertial energy-momentum PI<sup style='margin-left:-7px;'>μ naturally leads to the gauge theory of volume-preserving diffeomorphisms of a four-dimensional inner space. To analyse scattering in this theory, the gauge field is coupled to two Dirac fields with different masses. Based on a generalized LSZ reduction formula the S-matrix element for scattering of two Dirac particles in the gravitational limit and the corresponding scattering cross-section are calculated to leading order in perturbation theory. Taking the non-relativistic limit for one of the initial particles in the rest frame of the other the Rutherford-like cross-section of a non-relativistic particle scattering off an infinitely heavy scatterer calculated quantum mechanically in Newtonian gravity is recovered. This provides a non-trivial test of the gauge field theory of volume-preserving diffeomorphisms as a quantum theory of gravity.
基金supported by NNSF of China grant 11271252by RFDP of Higher Education of China grant 20110073110054by FP7-PEOPLE-2012-IRSES-316338 of Europe
文摘In this paper we first survey the results on the embedding flow problem of dif-feomorphisms in higher dimensional spaces. Next we present some new results on the characterization of semi-unipotent diffeomorphisms in R3, which have a formal embedding flows.
文摘Viewing gravitational energy-momentum as equal by observation, but different in essence from inertial energymomentum naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner Minkowski space which can describe gravitation at the classical level. This theory is quantized in the path integral formalism starting with a non-covariant Hamiltonian formulation with unconstrained canonical field variables and a manifestly positive Hamiltonian. The relevant path integral measure and weight are then brought into a Lorentz- and gauge-covariant form allowing to express correlation functions—applying the De Witt-Faddeev-Popov approach—in any meaningful gauge. Next the Feynman rules are developed and the quantum effective action at one loop in a background field approach is renormalized which results in an asymptotically free theory without presence of other fields and in a theory without asymptotic freedom including the Standard Model (SM) fields. Finally the BRST apparatus is developed as preparation for the renormalizability proof to all orders and a sketch of this proof is given.
基金Supported by FCT-Fundao para a Ciência e a Tecnologia and CNPq-Brazil(Grant No.PEst-OE/MAT/UI0212/2011)
文摘Let ACD（M, SL（d,R）） denote the pairs （f, A） so that f∈ A C Diff^1（M） is a C^1-Anosov transitive diffeomorphisms and A is an SL（d,R） cocycle dominated with respect to f. We prove that open and densely in ACD（M, SL（d,R））, in appropriate topologies, the pair （f,A） has simple spectrum with respect to the unique maximal entropy measure μf. Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in AUtLeb（M） × LP（M, SL（d, R））.
基金supported in part by the National Foundation for Science and Technology Development,Vietnam (Grant No.101.01.30.09)
文摘A suffcient condition for a set of calibrated submanifolds to be area-minimizing with multiplicities,also call weighted area-minimizing under diffeomorphisms (WAMD) is stated.We construct some WAMD submanifolds by assembling pieces of special Lagrangian (SL) normal bundles including the one of three surfaces meeting at an angle of 120° along soap-film-like singularities.We also mention a symmetry property of SL submanifolds and Bjrling type problem for SL normal bundles.
文摘We prove that the restrictions of the conjugacy between two Anosov diffeomorphisms of the twotorus to the stable and unstable manifolds are quasisymmetric homeomorphisms.
基金supported by the RFBR(Grant Nos.06-02-04012,08-01-00014)the program of Support of Scientific Schools(Grant No.NSH-3224.2008.1)Scientific Program of RAS"Nonlinear Dynamics"
文摘We discuss the twistor quantization problem for the classical system (V d ,A d ), represented by the phase space V d , identified with the Sobolev space H 0 1/2 (S 1,? d ) of half-differentiable vector functions on the circle, and the algebra of observables A d , identified with the semi-direct product of the Heisenberg algebra of V d and the algebra Vect(S 1) of tangent vector fields on the circle.
文摘Inertial and gravitational mass or energy momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The gauge-fixed action and the path integral measure occurring in the generating functional for the quantum Green functions of the theory are shown to obey a BRST-type symmetry. The related Zinn-Justin-type equation restricting the corresponding quantum effective action is established. This equation limits the infinite parts of the quantum effective action to have the same form as the gauge-fixed Lagrangian of the theory proving its spacetime renormalizability. The inner space integrals occurring in the quantum effective action which are divergent due to the gauge group’s infinite volume are shown to be regularizable in a way consistent with the symmetries of the theory demonstrating as a byproduct that viable quantum gauge field theories are not limited to finite-dimensional compact gauge groups as is commonly assumed.
基金Supported by the Natural Science Foundation of Tsinghua University.
文摘In this paper,we will use the embedding flows in[1],[2]to give a complete descrip- tion of the smooth centralizers and iterate radicals of all C^r(r≥2)Morse-Smale diffeomorphisms of the circle S^1.As a result,we prove that every centralizer is a solvable subgroup of Diff^r(S^1).
