The integrability of symplectic twist maps without conjugate points

2019 ◽  
Vol 41 (1) ◽  
pp. 48-65
Author(s):  
MARC ARCOSTANZO
Keyword(s):  
Cotangent Bundle ◽  
Conjugate Points ◽  
Dimensional Torus ◽  
Twist Map ◽  
Twist Maps

It is proved that a symplectic twist map of the cotangent bundle $T^{\ast }\mathbb{T}^{d}$ of the $d$-dimensional torus that is without conjugate points is $C^{0}$-integrable, that is  $T^{\ast }\mathbb{T}^{d}$ is foliated by a family of invariant $C^{0}$ Lagrangian graphs.

Rigidity of integrable twist maps and a theorem of Moser

1998 ◽  
Vol 18 (3) ◽  
pp. 725-730
Author(s):  
KARL FRIEDRICH SIBURG
Keyword(s):  
Higher Dimensions ◽  
Point Of View ◽  
Compact Set ◽  
Rigidity Result ◽  
Twist Map ◽  
Twist Maps

According to a theorem of Moser, every monotone twist map $\varphi$ on the cylinder ${\Bbb S}^1\times {\Bbb R}$, which is integrable outside a compact set, is the time-1-map $\varphi_H^1$ of a fibrewise convex Hamiltonian $H$. In this paper we prove that if this particular flow $\varphi_H^t$ is also integrable outside a compact set, then $\varphi$ has to be integrable on the whole cylinder (and vice versa, of course). From this dynamical point of view, integrable twist maps appear to be quite rigid.As is shown in the appendix, an analogous rigidity result becomes trivial in higher dimensions.

Propriétés des attracteurs de Birkhoff

1988 ◽  
Vol 8 (2) ◽  
pp. 241-310 ◽  
Author(s):  
P. Le Calvez
Keyword(s):  
Periodic Point ◽  
Infinite Number ◽  
Twist Map ◽  
Rotation Numbers ◽  
Twist Maps

AbstractWe study dissipative twist maps of the annulus, following the ideas of G. D. Birkhoff explained in an article of 1932.In the first part, we give complete and rigorous proofs of the results of this article. We define the Birkhoff attractor of a dissipative twist map which has an attracting bounded annulus, we give its main properties and we define its upper and lower rotation numbers.In the second part we give further results on these sets, thus we show that they often coincide with the closure of a hyperbolic periodic point and that they can contain an infinite number of sinks. We also show that the Birkhoff attractors don't depend on a continuous way on the maps.

Birkhoff-Hénon attractors for dissipative perturbations of area-preserving twist maps

1994 ◽  
Vol 14 (4) ◽  
pp. 807-815 ◽  
Author(s):  
Leonardo Mora
Keyword(s):  
Recent Result ◽  
Invariant Curve ◽  
Twist Map ◽  
Saddle Node ◽  
Twist Maps ◽  
Area Preserving

AbstractWe prove that an area-preserving twist map having an invariant curve, can be approximated by a twist map exhibiting a Birkhoff-Hénon attractor. This is done by showing that the invariant curve can be perturbed into a saddle-node cycle with criticalities and by using a recent result reported by Diaz, Rocha and Viana.

A method for proving that monotone twist maps have no invariant circles

1991 ◽  
Vol 11 (1) ◽  
pp. 79-84 ◽  
Author(s):  
Irwin Jungreis
Keyword(s):  
Existence Theorem ◽  
Computer Assisted ◽  
Twist Map ◽  
Standard Map ◽  
Rotation Numbers ◽  
Invariant Circles ◽  
Twist Maps ◽  
Parameter Values

AbstractWe present an existence theorem for certain kinds of orbits of a monotone twist map and use it to obtain a criterion for proving that there are no invariant circles with a certain range of rotation numbers. We have used this criterion to prove (computer assisted) that the standard map has no invariant circles for several parameter values includingk= 0.9718.

ELLIPTIC NON-BIRKHOFF PERIODIC ORBITS IN THE TWIST MAPS

1993 ◽  
Vol 03 (01) ◽  
pp. 165-185 ◽  
Author(s):  
ARTURO OLVERA ◽  
CARLES SIMÓ
Keyword(s):  
Periodic Orbits ◽  
Periodic Points ◽  
Irrational Rotation ◽  
Point Of View ◽  
Twist Map ◽  
Qualitative And Quantitative ◽  
Open Set ◽  
Twist Maps ◽  
Rotational Curve ◽  
Irrational Rotation Number

We consider a perturbed twist map when the perturbation is big enough to destroy the invariant rotational curve (IRC) with a given irrational rotation number. Then an invariant Cantorian set appears. From another point of view, the destruction of the IRC is associated with the appearance of heteroclinic connections between hyperbolic periodic points. Furthermore the destruction of the IRC is also associated with the existence of non-Birkhoff orbits. In this paper we relate the different approaches. In order to explain the creation of non-Birkhoff orbits, we provide qualitative and quantitative models. We show the existence of elliptic non-Birkhoff periodic orbits for an open set of values of the perturbative parameter. The bifurcations giving rise to the elliptic non-Birkhoff orbits and other related bifurcations are analysed. In the last section, we show a celestial mechanics example displaying the described behavior.

scholarly journals Lower and upper bounds for the Lyapunov exponents of twisting dynamics: a relationship between the exponents and the angle of Oseledets’ splitting

2012 ◽  
Vol 33 (3) ◽  
pp. 693-712 ◽  
Author(s):  
M.-C. ARNAUD
Keyword(s):  
Lyapunov Exponent ◽  
Lower Bound ◽  
Lyapunov Exponents ◽  
Cotangent Bundle ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Hamiltonian Flows ◽  
Twist Maps ◽  
The Mean ◽  
Minimizing Measures

AbstractWe consider locally minimizing measures for conservative twist maps of the $d$-dimensional annulus and for Tonelli Hamiltonian flows defined on a cotangent bundle $T^*M$. For weakly hyperbolic measures of such type (i.e. measures with no zero Lyapunov exponents), we prove that the mean distance/angle between the stable and unstable Oseledets bundles gives an upper bound on the sum of the positive Lyapunov exponents and a lower bound on the smallest positive Lyapunov exponent. We also prove some more precise results.

scholarly journals GAUGE-INVARIANT HAMILTONIAN FORMULATION OF LATTICE YANG-MILLS THEORY AND THE HEISENBERG DOUBLE

1995 ◽  
Vol 10 (34) ◽  
pp. 2619-2631 ◽  
Author(s):  
S.A. FROLOV
Keyword(s):  
Lie Group ◽  
Cotangent Bundle ◽  
Hamiltonian Formulation ◽  
Dimensional Torus ◽  
Gauge Invariant ◽  
Flat Connections ◽  
Yang Mills ◽  
Temporal Gauge ◽  
Heisenberg Double ◽  
Limiting Case

It is known that to get the usual Hamiltonian formulation of lattice Yang-Mills theory in the temporal gauge A0=0 one should place on each link a cotangent bundle of a Lie group. The cotangent bundle may be considered as a limiting case of a so-called Heisenberg double of a Lie group which is one of the basic objects in the theory of Lie-Poisson and quantum groups. It is shown in the paper that there is a generalization of the usual Hamiltonian formulation to the case of the Heisenberg double. The physical phase space of the (1+1)-dimensional γ-deformed Yang-Mills model is proved to be equivalent to the moduli space of flat connections on a two-dimensional torus.

scholarly journals Symplectic twist maps without conjugate points

2004 ◽  
Vol 141 (1) ◽  
pp. 235-247 ◽  
Author(s):  
M. L. Bialy ◽  
R. S. MacKay
Keyword(s):  
Conjugate Points ◽  
Twist Maps

scholarly journals A Topological Approach to Bend-Twist Maps with Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Anna Pascoletti ◽  
Fabio Zanolin
Keyword(s):  
Fixed Point Theorems ◽  
Small Perturbations ◽  
Topological Approach ◽  
Twist Map ◽  
Existence And Multiplicity ◽  
Twist Maps ◽  
Area Preserving Maps ◽  
Planar Systems ◽  
Twist Theorem ◽  
Area Preserving

In this paper we reconsider, in a purely topological framework, the concept of bend-twist map previously studied in the analytic setting by Tongren Ding in (2007). We obtain some results about the existence and multiplicity of fixed points which are related to the classical Poincaré-Birkhoff twist theorem for area-preserving maps of the annulus; however, in our approach, like in Ding (2007), we do not require measure-preserving conditions. This makes our theorems in principle applicable to nonconservative planar systems. Some of our results are also stable for small perturbations. Possible applications of the fixed point theorems for topological bend-twist maps are outlined in the last section.

scholarly journals Boundaries of instability zones for symplectic twist maps

2013 ◽  
Vol 13 (1) ◽  
pp. 19-41 ◽  
Author(s):  
M.-C. Arnaud
Keyword(s):  
Rotation Number ◽  
Irrational Rotation ◽  
Invariant Curve ◽  
Partial Answer ◽  
Twist Map ◽  
Instability Zone ◽  
Counter Example ◽  
Twist Maps ◽  
Irrational Rotation Number ◽  
The One

AbstractVery few things are known about the curves that are at the boundary of the instability zones of symplectic twist maps. It is known that in general they have an irrational rotation number and that they cannot be KAM curves. We address the following questions. Can they be very smooth? Can they be non-${C}^{1} $?Can they have a Diophantine or a Liouville rotation number? We give a partial answer for${C}^{1} $and${C}^{2} $twist maps.In Theorem 1, we construct a${C}^{2} $symplectic twist map$f$of the annulus that has an essential invariant curve$\Gamma $such that$\bullet $ $\Gamma $is not differentiable;$\bullet $the dynamics of${f}_{\vert \Gamma } $is conjugated to the one of a Denjoy counter-example;$\bullet $ $\Gamma $is at the boundary of an instability zone for$f$.Using the Hayashi connecting lemma, we prove in Theroem 2 that any symplectic twist map restricted to an essential invariant curve can be embedded as the dynamics along a boundary of an instability zone for some${C}^{1} $symplectic twist map.

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