Non-differentiable irrational curves for twist map

2021 ◽  
pp. 1-9
Author(s):  
ARTUR AVILA ◽  
BASSAM FAYAD
Keyword(s):  
Invariant Curve ◽  
Twist Map

Abstract We construct a $C^1$ symplectic twist map g of the annulus that has an essential invariant curve $\Gamma $ such that $\Gamma $ is not differentiable and g restricted to $\Gamma $ is minimal.

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.

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.

scholarly journals Generic pseudogroups on and the topology of leaves

2013 ◽  
Vol 149 (8) ◽  
pp. 1401-1430 ◽  
Author(s):  
J.-F. Mattei ◽  
J. C. Rebelo ◽  
H. Reis
Keyword(s):  
Invariant Curve ◽  
Holomorphic Foliation ◽  
Simply Connected

AbstractWe show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0\in \mathbb{C} $ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular, in the classical and simplest case arising from local nilpotent foliations possessing a unique separatrix which is given by a cusp of the form $\{ {y}^{2} - {x}^{2n+ 1} = 0\} $, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves.

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.

scholarly journals Twist sets for maps of the circle

1984 ◽  
Vol 4 (3) ◽  
pp. 391-404 ◽  
Author(s):  
Michał Misiurewicz
Keyword(s):  
Rotation Number ◽  
Closed Set ◽  
Twist Map ◽  
Rotation Interval ◽  
Continuous Map ◽  
The One

AbstractLet f be a continuous map of degree one of the circle onto itself. We prove that for every number a from the rotation interval of f there exists an invariant closed set A consisting of points with rotation number a and such that f restricted to A preserves the order. This result is analogous to the one in the case of a twist map of an annulus.

scholarly journals Stability, Neimark–Sacker Bifurcation, and Approximation of the Invariant Curve of Certain Homogeneous Second-Order Fractional Difference Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Mirela Garić-Demirović ◽  
Samra Moranjkić ◽  
Mehmed Nurkanović ◽  
Zehra Nurkanović
Keyword(s):  
Difference Equation ◽  
Equilibrium Point ◽  
Asymptotic Approximation ◽  
Second Order ◽  
Invariant Curve ◽  
Unique Equilibrium ◽  
Fractional Difference ◽  
Global Character ◽  
Fractional Difference Equation

We investigate the local and global character of the unique equilibrium point and boundedness of the solutions of certain homogeneous fractional difference equation with quadratic terms. Also, we consider Neimark–Sacker bifurcations and give the asymptotic approximation of the invariant curve.

scholarly journals On the Measure-Preserving Mappings with Three-Dimensions

1993 ◽  
Vol 132 ◽  
pp. 73-89
Author(s):  
Yi-Sui Sun
Keyword(s):  
Fixed Point ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Invariant Curve ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Kolmogorov Entropy ◽  
Numerical Exploration ◽  
Area Preserving ◽  
Dimensional Mapping

AbstractWe have systematically made the numerical exploration about the perturbation extension of area-preserving mappings to three-dimensional ones, in which the fixed points of area preserving are elliptic, parabolic or hyperbolic respectively. It has been observed that: (i) the invariant manifolds in the vicinity of the fixed point generally don’t exist (ii) when the invariant curve of original two-dimensional mapping exists the invariant tubes do also in the neighbourhood of the invariant curve (iii) for the perturbation extension of area-preserving mapping the invariant manifolds can only be generated in the subset of the invariant manifolds of original two-dimensional mapping, (iv) for the perturbation extension of area preserving mappings with hyperbolic or parabolic fixed point the ordered region near and far from the invariant curve will be destroyed by perturbation more easily than the other one, This is a result different from the case with the elliptic fixed point. In the latter the ordered region near invariant curve is solid. Some of the results have been demonstrated exactly.Finally we have discussed the Kolmogorov Entropy of the mappings and studied some applications.

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