scholarly journals Topology of optimal flows with collective dynamics on closed orientable surfaces

2020 ◽  
Vol 13 (2) ◽  
pp. 50-67
Author(s):  
Alexandr Olegovich Prishlyak ◽  
Mariya Viktorovna Loseva
Keyword(s):  
Morse Function ◽  
Hamiltonian Dynamics ◽  
Closed Surface ◽  
Topological Invariant ◽  
The Other ◽  
Heteroclinic Cycles ◽  
Collective Dynamics ◽  
Gradient Dynamics ◽  
Orientable Surfaces

We consider flows on a closed surface with one or more heteroclinic cycles that divide the surface into two regions. One of the region has gradient dynamics, like Morse fields. The other region has Hamiltonian dynamics generated by the field of the skew gradient of the simple Morse function. We construct the complete topological invariant of the flow using the Reeb and Oshemkov-Shark graphs and study its properties. We describe all possible structures of optimal flows with collective dynamics on oriented surfaces of genus no more than 2, both for flows containing a center and for flows without it.

Complete systems of surfaces in 3-manifolds

1997 ◽  
Vol 122 (1) ◽  
pp. 185-191 ◽  
Author(s):  
FENGCHUN LEI
Keyword(s):  
Finite Number ◽  
Boundary Component ◽  
Pairwise Disjoint ◽  
Complete System ◽  
Closed Surface ◽  
The Other ◽  
Closed Curves ◽  
Orientable Surfaces

A complete system (CS) [Jscr ]={J1, ..., Jn} on a connected closed surface F is a collection of pairwise disjoint simple closed curves on F such that the surface obtained by cutting F open along [Jscr ] is a 2-sphere with 2n-holes. Two CSs on F are equivalent if each can be obtained from the other via finite number of slides (defined in Section 1) and isotopies. Let M be a 3-manifold and F a boundary component of M of genus n. A CS of surfaces for M is a CS on F which bounds n pairwise disjoint incompressible orientable surfaces in M. When [Jscr ] is a CS of discs on the boundary of a handlebody V, it is well known that any CS on F which is equivalent to [Jscr ] is also a CS of discs for V. Our first result says that the same thing happens for a CS of surfaces for M, that is, if [Jscr ] is a CS of surfaces for M, then any CS equivalent to [Jscr ] is also a CS of surfaces for M. The following theorem is our main result on CS of surfaces in 3-manifolds:

Coupled systems of two-dimensional turbulence

2013 ◽  
Vol 732 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Hamiltonian System ◽  
Energy Conservation ◽  
Strong Correlation ◽  
Conservation Law ◽  
Hamiltonian Dynamics ◽  
Coupled Systems ◽  
The Other ◽  
Two Dimensional ◽  
Coherent Vortices ◽  
Fluid Particles

AbstractOrdinary two-dimensional turbulence corresponds to a Hamiltonian dynamics that conserves energy and the vorticity on fluid particles. This paper considers coupled systems of two-dimensional turbulence with three distinct governing dynamics. One is a Hamiltonian dynamics that conserves the vorticity on fluid particles and a quantity analogous to the energy that causes the system members to develop a strong correlation in velocity. The other two dynamics considered are non-Hamiltonian. One conserves the vorticity on particles but has no conservation law analogous to energy conservation; the other conserves energy and enstrophy but it does not conserve the vorticity on fluid particles. The coupled Hamiltonian system behaves like two-dimensional turbulence, even to the extent of forming isolated coherent vortices. The other two dynamics behave very differently, but the behaviours of all four dynamics are accurately predicted by the methods of equilibrium statistical mechanics.

scholarly journals Large amplitude oscillations for a class of symmetric polynomial differential systems in R³

2007 ◽  
Vol 79 (4) ◽  
pp. 563-575 ◽  
Author(s):  
Jaume Llibre ◽  
Marcelo Messias
Keyword(s):  
Large Amplitude ◽  
Symmetric Polynomial ◽  
The Other ◽  
Poincaré Compactification ◽  
Heteroclinic Loop ◽  
Differential Systems ◽  
Heteroclinic Cycles ◽  
Polynomial Differential Systems ◽  
Global Study ◽  
Straight Lines

In this paper we study a class of symmetric polynomial differential systems in R³, which has a set of parallel invariant straight lines, forming degenerate heteroclinic cycles, which have their two singular endpoints at infinity. The global study near infinity is performed using the Poincaré compactification. We prove that for all n <FONT FACE=Symbol>Î</FONT> N there is epsilonn > 0 such that for 0 < epsilon < epsilonn the system has at least n large amplitude periodic orbits bifurcating from the heteroclinic loop formed by the two invariant straight lines closest to the x-axis, one contained in the half-space y > 0 and the other in y < 0.

scholarly journals The Geometric Theory of the Fundamental Germ

2008 ◽  
Vol 190 ◽  
pp. 1-34 ◽  
Author(s):  
T. M. Gendron
Keyword(s):  
Group Actions ◽  
Geometric Theory ◽  
Universal Cover ◽  
Closed Surface ◽  
Topological Invariant ◽  
Hyperbolic Type ◽  
Smooth Structure

The fundamental germ is a generalization of π1, first defined for laminations which arise through group actions [4]. In this paper, the fundamental germ is extended to any lamination having a dense leaf admitting a smooth structure. In addition, an amplification of the fundamental germ called the mother germ is constructed, which is, unlike the fundamental germ, a topological invariant. The fundamental germs of the antenna lamination and the PSL(2,ℤ) lamination are calculated, laminations for which the definition in [4] was not available. The mother germ is used to give a new proof of a Nielsen theorem for the algebraic universal cover of a closed surface of hyperbolic type.

scholarly journals REDUCING DEHN FILLINGS AND SMALL SURFACES

2005 ◽  
Vol 92 (1) ◽  
pp. 203-223 ◽  
Author(s):  
SANGYOP LEE ◽  
SEUNGSANG OH ◽  
MASAKAZU TERAGAITO
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Hyperbolic Manifolds ◽  
The Other ◽  
Möbius Band ◽  
Dehn Fillings ◽  
Orientable Surfaces

In this paper we investigate the distances between Dehn fillings on a hyperbolic 3-manifold that yield 3-manifolds containing essential small surfaces including non-orientable surfaces. In particular, we study the situations where one filling creates an essential sphere or projective plane, and the other creates an essential sphere, projective plane, annulus, Möbius band, torus or Klein bottle, for all eleven pairs of such non-hyperbolic manifolds.

scholarly journals Minimal Penner dilatations on nonorientable surfaces

2019 ◽  
pp. 1-32 ◽  
Author(s):  
Livio Liechti ◽  
Balázs Strenner
Keyword(s):  
Accumulation Point ◽  
Closed Surface ◽  
Alexander Polynomials ◽  
Nonorientable Surfaces ◽  
Accumulation Points ◽  
Mapping Classes ◽  
Key Techniques ◽  
Fibered Links ◽  
Orientable Surfaces ◽  
Skein Relation

For any nonorientable closed surface, we determine the minimal dilatation among pseudo-Anosov mapping classes arising from Penner’s construction. We deduce that the sequence of minimal Penner dilatations has exactly two accumulation points, in contrast to the case of orientable surfaces where there is only one accumulation point. One of our key techniques is representing pseudo-Anosov dilatations as roots of Alexander polynomials of fibered links and comparing dilatations using the skein relation for Alexander polynomials.

scholarly journals Exploring the “Rubik's Magic” Universe

2017 ◽  
Vol 4 (7) ◽  
pp. 29-64
Author(s):  
Maurizio Paolini
Keyword(s):  
Topological Type ◽  
Topological Invariant ◽  
The Other ◽  
Coordinate Plane ◽  
Common Plane ◽  
3D Shapes ◽  
The Universe

Abstract By using two different invariants for the Rubik’s Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of constructible configurations of the puzzle. Finding the set of actually constructible shapes remains however a challenging task, that we tackle by first reducing the target shapes to specific configurations: the octominoid 3D shapes, with all tiles parallel to one coordinate plane; and the planar “face-up” shapes, with all tiles (considered of infinitesimal width) lying in a common plane and without superposed consecutive tiles. There are still plenty of interesting configurations that do not belong to either of these two collections. The set of constructible configurations (those that can be obtained by manipulation of the undecorated puzzle from the starting situation) is a subset of the set of configurations with vanishing invariants. We were able to actually construct all octominoid shapes with vanishing invariants and most of the planar “face-up” configurations. Particularly important is the topological invariant, of which we recently found mention in [7] by Tom Verhoeff.

WITTEN LAPLACIAN ON PINNED PATH GROUP AND ITS EXPECTED SEMICLASSICAL BEHAVIOR

2003 ◽  
Vol 06 (supp01) ◽  
pp. 103-114 ◽  
Author(s):  
SHIGEKI AIDA
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Critical Points ◽  
Energy Function ◽  
Morse Function ◽  
Riemannian Metric ◽  
The Other ◽  
Compact Lie Group ◽  
Semiclassical Analysis ◽  
Witten Laplacian

A right invariant Riemannian metric is defined on a pinned path group over a compact Lie group G. The energy function of the path is a Morse function and the critical points are geodesics. We calculate the eigenvalues of the Hessian at the critical points when G=SU(n). On the other hand, there exists a pinned Brownian motion measure νλ with a variance parameter 1/λ on the pinned path group and we can define a Hodge-Kodaira-Witten type operator □λ on L2(νλ)-space of p-forms on the pinned path group. By using the explicit expression of eigenvalues of the Hessian of the energy function, we discuss the asymptotic behavior of the botton of the spectrum of □λ as λ→∞ by a formal semiclassical analysis.

On certain Subgroups of the fundamental group of a closed surface

1970 ◽  
Vol 67 (1) ◽  
pp. 17-18 ◽  
Author(s):  
William Jaco
Keyword(s):  
Fundamental Group ◽  
Closed Surface ◽  
The Other

1. Introduction. Although Corollaries 4 and 5 to Theorem 1 below appear elsewhere in the literature (1, 2), the proofs given seem to use rather long and involved arguments and refer to other results in the literature for their completeness. The proofs given below are brief and follow quite naturally in sequence with the other corollaries from Theorem 1. The arguments presented are independent of references to the literature except for the reference in the proof of Theorem 1 to Lemma 2·1 of (3), which is well known to topologists. For our purposes this theorem may be stated as: For each open 2-manifold M (non-compact and without boundary), there is a subcomplex L made up of edges of some triangulation of M such that the open simplicial neighbourhood of L is piecewise linearly homeomorphic to M.

scholarly journals Flows with collective dynamics on a sphere

2021 ◽  
Vol 14 (1) ◽  
pp. 60-79
Author(s):  
Андрій Прус ◽  
Олександр Пришляк ◽  
Софія Гурака
Keyword(s):  
Topological Invariant ◽  
Collective Dynamics ◽  
Compact Surfaces

In this article different properties of flow codes are studied and a diagram is constructed as a whole topological invariant of them. In particular, flows with no more than 6 saddles are described. Two types of simple bifurcations: positive and negative – are considered as well. Summarizing the results on compact surfaces with boundary remains an interesting question for future works.  

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