scholarly journals Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour

1982 ◽  
Vol 2 (3-4) ◽  
pp. 439-463 ◽  
Author(s):  
Feliks Przytycki
Keyword(s):  
Fixed Point ◽  
Lebesgue Measure ◽  
Bifurcation Parameter ◽  
Two Dimensional ◽  
Open Area ◽  
Dimensional Torus ◽  
Characteristic Exponents ◽  
Almost Everywhere ◽  
Lyapunov Characteristic Exponents ◽  
Real Analytic

AbstractWe find very simple examples of C∞-arcs of diffeomorphisms of the two-dimensional torus, preserving the Lebesgue measure and having the following properties: (1) the beginning of an arc is inside the set of Anosov diffeomorphisms; (2) after the bifurcation parameter every diffeomorphism has an elliptic fixed point with the first Birkhoff invariant non-zero (the KAM situation) and an invariant open area with almost everywhere non-zero Lyapunov characteristic exponents, moreover where the diffeomorphism has Bernoulli property; (3) the arc is real-analytic except on two circles (for each value of parameter) which are inside the Bernoulli property area.

A correction to ‘Analyticity and metric transitivity on the torus’

1999 ◽  
Vol 19 (1) ◽  
pp. 259-261
Author(s):  
SOL SCHWARTZMAN
Keyword(s):  
Differential Equations ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Topological Transitivity ◽  
Topological Disc ◽  
Real Analytic

In [2], flows on the standard two-dimensional torus given by the differential equations \begin{equation*} \frac{dx}{dt}=a-Fy(x,y),\quad \frac{dv}{dt}=b+Fx(x,y) \end{equation*} were considered. It was assumed that $F(x,y)$ was real analytic and of period one in both $x$ and $y$. A key step in proving the results in [2] was to show that one could conclude topological transitivity for the flow provided one assumed: \begin{enumerate} \item[(a)] $a/b$ is irrational; \item[(b)] there does not exist a topological disc on the torus that is invariant under the flow. \end{enumerate}

Rotation measures for homeomorphisms of the torus homotopic to a Dehn twist

1997 ◽  
Vol 17 (3) ◽  
pp. 575-591 ◽  
Author(s):  
H. ERIK DOEFF
Keyword(s):  
Fixed Point ◽  
Rotation Number ◽  
Rational Number ◽  
Dehn Twist ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Rotation Vectors ◽  
Rotation Set ◽  
Area Preserving

We extend the theory of rotation vectors to homeomorphisms of the two-dimensional torus that are homotopic to a Dehn twist. We define a one-dimensional rotation number and recreate the theory of the homotopic case to the identity case. We prove that if such a map is area preserving and has mean rotation number zero, then it must have a fixed point. We prove that the rotation set is a compact interval, and that if the rotation interval contains two distinct numbers, then for any rational number in the rotation set there exists a periodic point with that rotation number. Finally, we prove that any interval with rational endpoints can be realized as the rotation set of a map homotopic to a Dehn twist.

scholarly journals Non-standard real-analytic realizations of some rotations of the circle

2015 ◽  
Vol 37 (5) ◽  
pp. 1369-1386 ◽  
Author(s):  
SHILPAK BANERJEE
Keyword(s):  
Two Dimensional ◽  
Dimensional Torus ◽  
Smooth Approximation ◽  
Analytic Set ◽  
Irrational Rotations ◽  
Zero Entropy ◽  
Real Analytic ◽  
Set Up ◽  
Uniquely Ergodic ◽  
Conjugation Method

We extend some aspects of the smooth approximation by conjugation method to the real-analytic set-up, and create examples of zero entropy, uniquely ergodic, real-analytic diffeomorphisms of the two-dimensional torus that are metrically isomorphic to some (Liouvillian) irrational rotations of the circle.

On the reducibility of two-dimensional linear quasi-periodic systems with small parameter

2014 ◽  
Vol 35 (7) ◽  
pp. 2334-2352 ◽  
Author(s):  
JUNXIANG XU ◽  
XUEZHU LU
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Lebesgue Measure ◽  
Periodic System ◽  
Coefficient Matrix ◽  
Periodic Systems ◽  
Two Dimensional ◽  
Constant Matrix ◽  
Small Parameters ◽  
Real Analytic

In this paper we consider a linear real analytic quasi-periodic system of two differential equations, whose coefficient matrix analytically depends on a small parameter and closes to constant. Under some non-resonance conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy assumption of the small parameter, we prove that the system is reducible for most of the sufficiently small parameters in the sense of the Lebesgue measure.

scholarly journals Bernoulli diffeomorphisms with n − 1 non-zero exponents

1981 ◽  
Vol 1 (1) ◽  
pp. 1-7 ◽  
Author(s):  
M. Brin
Keyword(s):  
Lebesgue Measure ◽  
Skew Product ◽  
Anosov Flow ◽  
Characteristic Exponents ◽  
Zero Characteristic ◽  
Almost Everywhere

AbstractFor every manifold of dimension n ≥ 5 a diffeomorphism f which has n − 1 non-zero characteristic exponents almost everywhere is constructed. The diffeomorphism preserves the Lebesgue measure and is Bernoulli with respect to this measure. To produce this example a diffeomorphism of the 2-disk is extended by means of an Anosov flow, and this skew product is embedded in ℝn.

scholarly journals Circular quiver gauge theories, isomonodromic deformations and $$W_N$$ fermions on the torus

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Giulio Bonelli ◽  
Fabrizio Del Monte ◽  
Pavlo Gavrylenko ◽  
Alessandro Tanzini
Keyword(s):  
Integrable System ◽  
Gauge Theories ◽  
The Self ◽  
Free Fermion ◽  
Two Dimensional ◽  
Isomonodromic Deformation ◽  
Specific Time ◽  
Dimensional Torus ◽  
Isomonodromic Deformations ◽  
Deformation Problem

AbstractWe study the relation between class $$\mathcal {S}$$ S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two-dimensional torus with punctures. Turning on the self-dual $$\Omega $$ Ω -background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding $$\tau $$ τ -function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to $$W_N$$ W N free fermion correlators on the torus.

TERBIUM MOLYBDATE: GROUP THEORY AND FLUCTUATIONS

1987 ◽  
Vol 01 (05n06) ◽  
pp. 239-244
Author(s):  
SERGE GALAM
Keyword(s):  
Fixed Point ◽  
Group Theory ◽  
Parameter Space ◽  
Landau Theory ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Theory Analysis ◽  
Continuous Transition ◽  
Dimensional Parameter ◽  
First Order

A new mechanism to explain the first order ferroelastic—ferroelectric transition in Terbium Molybdate (TMO) is presented. From group theory analysis it is shown that in the two-dimensional parameter space ordering along either an axis or a diagonal is forbidden. These symmetry-imposed singularities are found to make the unique stable fixed point not accessible for TMO. A continuous transition even if allowed within Landau theory is thus impossible once fluctuations are included. The TMO transition is therefore always first order. This explanation is supported by experimental results.

scholarly journals Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

2018 ◽  
Vol 28 (04) ◽  
pp. 1830011
Author(s):  
Mio Kobayashi ◽  
Tetsuya Yoshinaga
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Chaotic Behavior ◽  
Periodic Points ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Bifurcation Structure ◽  
Unstable Fixed Point ◽  
Gaussian Map ◽  
Gaussian Maps

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the [Formula: see text]-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

scholarly journals Stable Arcs Connecting Polar Cascades on a Torus

2021 ◽  
Vol 17 (1) ◽  
pp. 23-37
Author(s):  
O. V. Pochinka ◽  
◽  
E. V. Nozdrinova ◽  
Keyword(s):  
Two Dimensional ◽  
Dimensional Torus ◽  
Positive Orientation ◽  
Stable Isotopic ◽  
Orientation Type

In the article, the components of the stable isotopic connection of polar gradient-like diffeomorphisms on a two-dimensional torus are found under the assumption that all non-wandering points are fixed and have a positive orientation type.

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