scholarly journals Volume growth in the component of fibered twists

2018 ◽  
Vol 20 (08) ◽  
pp. 1850014 ◽  
Author(s):  
Joontae Kim ◽  
Myeonggi Kwon ◽  
Junyoung Lee
Keyword(s):  
Lower Bound ◽  
Symplectic Manifold ◽  
Infinite Order ◽  
Floer Homology ◽  
Volume Growth ◽  
Spectral Sequences ◽  
Connected Component ◽  
Text Type ◽  
Infinite Dimensional ◽  
Homology Groups

For a Liouville domain [Formula: see text] whose boundary admits a periodic Reeb flow, we can consider the connected component [Formula: see text] of fibered twists. In this paper, we investigate an entropy-type invariant, called the slow volume growth, in the component [Formula: see text] and give a uniform lower bound of the growth using wrapped Floer homology. We also show that [Formula: see text] has infinite order in [Formula: see text] if there is an admissible Lagrangian [Formula: see text] in [Formula: see text] whose wrapped Floer homology is infinite dimensional. We apply our results to fibered twists coming from the Milnor fibers of [Formula: see text]-type singularities and complements of a symplectic hypersurface in a real symplectic manifold. They admit so-called real Lagrangians, and we can explicitly compute wrapped Floer homology groups using a version of Morse–Bott spectral sequences.

scholarly journals FLOER HOMOLOGY FOR SYMPLECTOMORPHISM

2009 ◽  
Vol 11 (06) ◽  
pp. 895-936 ◽  
Author(s):  
HAI-LONG HER
Keyword(s):  
Fixed Points ◽  
Symplectic Manifold ◽  
Floer Homology ◽  
Symplectic Manifolds ◽  
Homology Groups ◽  
Symplectic Diffeomorphism ◽  
Novikov Ring ◽  
Compact Symplectic Manifold

Let (M,ω) be a compact symplectic manifold, and ϕ be a symplectic diffeomorphism on M, we define a Floer-type homology FH*(ϕ) which is a generalization of Floer homology for symplectic fixed points defined by Dostoglou and Salamon for monotone symplectic manifolds. These homology groups are modules over a suitable Novikov ring and depend only on ϕ up to a Hamiltonian isotopy.

A note on the algebra of Poisson brackets

1984 ◽  
Vol 96 (1) ◽  
pp. 45-60 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Poisson Brackets ◽  
Infinite Dimensional

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

scholarly journals Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree

2016 ◽  
Vol 9 (2) ◽  
pp. 607-686
Author(s):  
Robert Lipshitz ◽  
Peter S. Ozsváth ◽  
Dylan P. Thurston
Keyword(s):  
Spectral Sequence ◽  
Floer Homology ◽  
Double Cover ◽  
Spectral Sequences

scholarly journals Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems

2015 ◽  
Vol 07 (03) ◽  
pp. 407-451 ◽  
Author(s):  
Urs Frauenfelder ◽  
Clémence Labrousse ◽  
Felix Schlenk
Keyword(s):  
Lower Bound ◽  
Polynomial Growth ◽  
Polynomial Complexity ◽  
Cohomology Ring ◽  
Volume Growth ◽  
Universal Cover ◽  
Geodesic Flows ◽  
Dynamical Complexity ◽  
Reeb Flows ◽  
Rank One

We give a uniform lower bound for the polynomial complexity of Reeb flows on the spherization (S*M, ξ) over a closed manifold. Our measure for the dynamical complexity of Reeb flows is slow volume growth, a polynomial version of topological entropy, and our lower bound is in terms of the polynomial growth of the homology of the based loop space of M. As an application, we extend the Bott–Samelson theorem from geodesic flows to Reeb flows: If (S*M, ξ) admits a periodic Reeb flow, or, more generally, if there exists a positive Legendrian loop of a fiber [Formula: see text], then M is a circle or the fundamental group of M is finite and the integral cohomology ring of the universal cover of M agrees with that of a compact rank one symmetric space.

scholarly journals COMPLEMENTARITY IN GENERIC OPEN QUANTUM SYSTEMS

2010 ◽  
Vol 24 (24) ◽  
pp. 2485-2509 ◽  
Author(s):  
SUBHASHISH BANERJEE ◽  
R. SRIKANTH
Keyword(s):  
Lower Bound ◽  
Uniform Distribution ◽  
Open Quantum Systems ◽  
Quantum Systems ◽  
Atomic Systems ◽  
Information Theoretic ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The Difference ◽  
Positive Operator Valued Measure

We develop a unified, information theoretic interpretation of the number-phase complementarity that is applicable both to finite-dimensional (atomic) and infinite-dimensional (oscillator) systems, with number treated as a discrete Hermitian observable and phase as a continuous positive operator valued measure (POVM). The relevant uncertainty principle is obtained as a lower bound on entropy excess, X, the difference between the entropy of one variable, typically the number, and the knowledge of its complementary variable, typically the phase, where knowledge of a variable is defined as its relative entropy with respect to the uniform distribution. In the case of finite-dimensional systems, a weighting of phase knowledge by a factor μ (> 1) is necessary in order to make the bound tight, essentially on account of the POVM nature of phase as defined here. Numerical and analytical evidence suggests that μ tends to 1 as the system dimension becomes infinite. We study the effect of non-dissipative and dissipative noise on these complementary variables for an oscillator as well as atomic systems.

Statistical mechanics of the melting transition in lattice models of polymers

1974 ◽  
Vol 337 (1611) ◽  
pp. 569-589 ◽  
Keyword(s):  
Infinite Order ◽  
Lattice Models ◽  
Melting Transition ◽  
Excluded Volume ◽  
Approximate Theory ◽  
Dimensional Theory ◽  
Infinite Dimensional ◽  
First Order ◽  
Exactly Solvable ◽  
Cross Links

Five two-dimensional lattice models, four with rotational isomeric and excluded volume interactions and one with cross links, are used to discuss the theory of the melting transition in polymers. The models have been chosen because they are isomorphic to exactly solvable six vertex and dimer models. The orders of the thermodynamic transitions are extremely varied from model to model, including first-order, 3/2 order and infinite order transitions. These models are used to test and reveal the shortcomings of the Flory–Huggins approximate theory, which is most aptly described as an infinite dimensional theory.

scholarly journals Deformation of Brody curves and mean dimension

2009 ◽  
Vol 29 (5) ◽  
pp. 1641-1657 ◽  
Author(s):  
MASAKI TSUKAMOTO
Keyword(s):  
Lower Bound ◽  
Projective Space ◽  
Parameter Space ◽  
Complex Plane ◽  
Deformation Theory ◽  
Holomorphic Map ◽  
Infinite Dimensional ◽  
Mean Dimension ◽  
The Mean

AbstractThe main purpose of this paper is to show that ideas of deformation theory can be applied to ‘infinite-dimensional geometry’. We develop the deformation theory of Brody curves. A Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite-dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves.

scholarly journals Brownian motion on stationary random manifolds

2016 ◽  
Vol 16 (02) ◽  
pp. 1660001
Author(s):  
Pablo Lessa
Keyword(s):  
Brownian Motion ◽  
Harmonic Functions ◽  
Dimensional Space ◽  
Volume Growth ◽  
Upper And Lower Bounds ◽  
Entropy Theory ◽  
Liouville Property ◽  
Average Volume ◽  
Infinite Dimensional ◽  
Bounded Harmonic Functions

We introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the entropy of an ergodic stationary random manifold is zero if and only if the manifold satisfies the Liouville property almost surely, and is positive if and only if it admits an infinite dimensional space of bounded harmonic functions almost surely. Upper and lower bounds for the entropy are provided in terms of the linear drift of Brownian motion and average volume growth of the manifold. Other almost sure properties of these random manifolds are also studied.

Connectivity Results of Complete Cubic Networks as Associated with Linearly Many Faults

2015 ◽  
Vol 15 (01n02) ◽  
pp. 1550007 ◽  
Author(s):  
EDDIE CHENG ◽  
KE QIU ◽  
ZHIZHANG SHEN
Keyword(s):  
Fault Tolerance ◽  
Lower Bound ◽  
Network Structure ◽  
Diagnostic Process ◽  
Connected Component ◽  
Vertex Connectivity ◽  
Number Of Components ◽  
Large Connected Component ◽  
The Common ◽  
Diagnosis Model

We propose the complete cubic network structure to extend the existing class of hierarchical cubic networks, and establish a general connectivity result which states that the surviving graph of a complete cubic network, when a linear number of vertices are removed, consists of a large (connected) component and a number of smaller components which altogether contain a limited number of vertices. As applications, we characterize several fault-tolerance properties for the complete cubic network, including its restricted connectivity, i.e., the size of a minimum vertex cut such that the degree of every vertex in the surviving graph has a guaranteed lower bound; its cyclic vertex-connectivity, i.e., the size of a minimum vertex cut such that at least two components in the surviving graph contain a cycle; its component connectivity, i.e., the size of a minimum vertex cut whose removal leads to a certain number of components in its surviving graph; and its conditional diagnosability, i.e., the maximum number of faulty vertices that can be detected via a self-diagnostic process, in terms of the common Comparison Diagnosis model.

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